Volume 196

International Series in Operations Research & Management Science

Series Editor

Frederick S. Hillier

Stanford University, CA, USA

For further volumes: http://​www.​springer.​com/​series/​6161

Robert J. Vanderbei

Linear ProgrammingFoundations and ExtensionsFourth Edition

Robert J. Vanderbei

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

ISSN 0884-8289e-ISSN 2214-7934

International Series in Operations Research & Management Science

ISBN 978-1-4614-7629-0e-ISBN 978-1-4614-7630-6

https://doi.org/10.1007/978-1-4614-7630-6

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Printed on acid-free paper

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To Krisadee,

Marisa and Diana

Preface

This book is about constrained optimization. It begins with a thorough treatment of linear programming and proceeds to convex analysis, network flows, integer programming, quadratic programming, and convex optimization. Along the way, dynamic programming and the linear complementarity problem are touched on as well.

The book aims to be a first introduction to the subject. Specific examples and concrete algorithms precede more abstract topics. Nevertheless, topics covered are developed in some depth, a large number of numerical examples are worked out in detail, and many recent topics are included, most notably interior-point methods. The exercises at the end of each chapter both illustrate the theory and, in some cases, extend it.

Prerequisites. The book is divided into four parts. The first two parts assume a background only in linear algebra. For the last two parts, some knowledge of multivariate calculus is necessary. In particular, the student should know how to use Lagrange multipliers to solve simple calculus problems in 2 and 3 dimensions.

Associated software. It is good to be able to solve small problems by hand, but the problems one encounters in practice are large, requiring a computer for their solution. Therefore, to fully appreciate the subject, one needs to solve large (practical) problems on a computer. An important feature of this book is that it comes with software implementing the major algorithms described herein. At the time of writing, software for the following five algorithms is available:

The two-phase simplex method as shown in Figure 6.​1.

The self-dual simplex method as shown in Figure 7.​1.

The path-following method as shown in Figure 18.​1.

The homogeneous self-dual method as shown in Figure 22.​1.

The long-step homogeneous self-dual method as described in Exercise 22.4.

The programs that implement these algorithms are written in C and can be easily compiled on most hardware platforms. Students/instructors are encouraged to install and compile these programs on their local hardware. Great pains have been taken to make the source code for these programs readable (see Appendix A). In particular, the names of the variables in the programs are consistent with the notation of this book.

There are two ways to run these programs. The first is to prepare the input in a standard computer-file format, called MPS format, and to run the program using such a file as input. The advantage of this input format is that there is an archive of problems stored in this format, called the NETLIB suite, that one can download and use immediately (a link to the NETLIB suite can be found at the web site mentioned below). But, this format is somewhat archaic and, in particular, it is not easy to create these files by hand. Therefore, the programs can also be run from within a problem modeling system called AMPL. AMPL allows one to describe mathematical programming problems using an easy to read, yet concise, algebraic notation. To run the programs within AMPL, one simply tells AMPL the name of the solver-program before asking that a problem be solved. The text that describes AMPL, Fourer et al. (1993) makes an excellent companion to this book. It includes a discussion of many practical linear programming problems. It also has lots of exercises to hone the modeling skills of the student.

Several interesting computer projects can be suggested. Here are a few suggestions regarding the simplex codes:

Incorporate the partial pricing strategy (see Section 8.7) into the two-phase simplex method and compare it with full pricing.

Incorporate the steepest-edge pivot rule (see Section 8.8) into the two-phase simplex method and compare it with the largest-coefficient rule.

Modify the code for either variant of the simplex method so that it can treat bounds and ranges implicitly (see Chapter 9), and compare the performance with the explicit treatment of the supplied codes.

Implement a “warm-start” capability so that the sensitivity analyses discussed in Chapter 7 can be done.

Extend the simplex codes to be able to handle integer programming problems using the branch-and-bound method described in Chapter 23.

As for the interior-point codes, one could try some of the following projects:

Modify the code for the path-following algorithm so that it implements the affine-scaling method (see Chapter 21), and then compare the two methods.

Modify the code for the path-following method so that it can treat bounds and ranges implicitly (see Section 20.3), and compare the performance against the explicit treatment in the given code.

Modify the code for the path-following method to implement the higher-order method described in Exercise 18.5. Compare.

Extend the path-following code to solve quadratic programming problems using the algorithm shown in Figure 24.​3.

Further extend the code so that it can solve convex optimization problems using the algorithm shown in Figure 25.2.

And, perhaps the most interesting project of all:

Compare the simplex codes against the interior-point code and decide for yourself which algorithm is better on specific families of problems.

The software implementing the various algorithms was developed using consistent data structures and so making fair comparisons should be straightforward. The software can be downloaded from the following web site:

http://​www.​princeton.​edu/​~rvdb/​LPbook/​

If, in the future, further codes relating to this text are developed (for example, a self-dual network simplex code), they will be made available through this web site.

Features. Here are some other features that distinguish this book from others:

The development of the simplex method leads to Dantzig’s parametric self-dual method. A randomized variant of this method is shown to be immune to the travails of degeneracy.

The book gives a balanced treatment to both the traditional simplex method and the newer interior-point methods. The notation and analysis is developed to be consistent across the methods. As a result, the self-dual simplex method emerges as the variant of the simplex method with most connections to interior-point methods.

From the beginning and consistently throughout the book, linear programming problems are formulated in symmetric form. By highlighting symmetry throughout, it is hoped that the reader will more fully understand and appreciate duality theory.

By slightly changing the right-hand side in the Klee–Minty problem, we are able to write down an explicit dictionary for each vertex of the Klee–Minty problem and thereby uncover (as a homework problem) a simple, elegant argument why the Klee-Minty problem requires 2 n  − 1 pivots to solve.

The chapter on regression includes an analysis of the expected number of pivots required by the self-dual variant of the simplex method. This analysis is supported by an empirical study.

There is an extensive treatment of modern interior-point methods, including the primal–dual method, the affine-scaling method, and the self-dual path-following method.

In addition to the traditional applications, which come mostly from business and economics, the book features other important applications such as the optimal design of truss-like structures and L ¹-regression.

Exercises on the Web. There is always a need for fresh exercises. Hence, I have created and plan to maintain a growing archive of exercises specifically created for use in conjunction with this book. This archive is accessible from the book’s web site:

http://​www.​princeton.​edu/​~rvdb/​LPbook/​

The problems in the archive are arranged according to the chapters of this book and use notation consistent with that developed herein.

Advice on solving the exercises. Some problems are routine while others are fairly challenging. Answers to some of the problems are given at the back of the book. In general, the advice given to me by Leonard Gross (when I was a student) should help even on the hard problems: follow your nose.

Audience. This book evolved from lecture notes developed for my introductory graduate course in linear programming as well as my upper-level undergraduate course. A reasonable undergraduate syllabus would cover essentially all of Part I (Simplex Method and Duality), the first two chapters of Part II (Network Flows and Applications), and the first chapter of Part IV (Integer Programming). At the graduate level, the syllabus should depend on the preparation of the students. For a well-prepared class, one could cover the material in Parts I and II fairly quickly and then spend more time on Parts III (Interior-Point Methods) and IV (Extensions).

Dependencies. In general, Parts II and III are completely independent of each other. Both depend, however, on the material in Part I. The first Chapter in Part IV (Integer Programming) depends only on material from Part I, whereas the remaining chapters build on Part III material.

Acknowledgments. My interest in linear programming was sparked by Robert Garfinkel when we shared an office at Bell Labs. I would like to thank him for his constant encouragement, advice, and support. This book benefited greatly from the thoughtful comments and suggestions of David Bernstein and Michael Todd. I would also like to thank the following colleagues for their help: Ronny Ben-Tal, Leslie Hall, Yoshi Ikura, Victor Klee, Irvin Lustig, Avi Mandelbaum, Marc Meketon, Narcis Nabona, James Orlin, Andrzej Ruszczynski, and Henry Wolkowicz. I would like to thank Gary Folven at Kluwer and Fred Hillier, the series editor, for encouraging me to undertake this project. I would like to thank my students for finding many typos and occasionally more serious errors: John Gilmartin, Jacinta Warnie, Stephen Woolbert, Lucia Wu, and Bing Yang.  My thanks to Erhan Çınlar for the many times he offered advice on questions of style. I hope this book reflects positively on his advice. Finally, I would like to acknowledge the support of the National Science Foundation and the Air Force Office of Scientific Research for supporting me while writing this book. In a time of declining resources, I am especially grateful for their support.

Robert J. Vanderbei

Princeton, NJ, USA

Preface to 2nd Edition

For the 2nd edition, many new exercises have been added. Also I have worked hard to develop online tools to aid in learning the simplex method and duality theory. These online tools can be found on the book’s web page: http://​www.​princeton.​edu/​~rvdb/​LPbook/​ and are mentioned at appropriate places in the text. Besides the learning tools, I have created several online exercises. These exercises use randomly generated problems and therefore represent a virtually unlimited collection of “routine” exercises that can be used to test basic understanding. Pointers to these online exercises are included in the exercises sections at appropriate points.

Some other notable changes include:

The chapter on network flows has been completely rewritten. Hopefully, the new version is an improvement on the original.

Two different fonts are now used to distinguish between the set of basic indices and the basis matrix.

The first edition placed great emphasis on the symmetry between the primal and the dual (the negative transpose property). The second edition carries this further with a discussion of the relationship between the basic and nonbasic matrices B and N as they appear in the primal and in the dual. We show that, even though these matrices differ (they even have different dimensions), B  − 1 N in the dual is the negative transpose of the corresponding matrix in the primal.

In the chapters devoted to the simplex method in matrix notation, the collection of variables $$z_{1},z_{2},\ldots ,z_{n},y_{1},y_{2},\ldots ,y_{m}$$ was replaced, in the first edition, with the single array of variables $$y_{1},y_{2},\ldots ,y_{n+m}$$. This caused great confusion as the variable y i in the original notation was changed to y n + i in the new notation. For the second edition, I have changed the notation for the single array to $$z_{1},z_{2},\ldots ,z_{n+m}$$.

A number of figures have been added to the chapters on convex analysis and on network flow problems.

The algorithm refered to as the primal–dual simplex method in the first edition has been renamed the parametric self-dual simplex method in accordance with prior standard usage.

The last chapter, on convex optimization, has been extended with a discussion of merit functions and their use in shortenning steps to make some otherwise nonconvergent problems converge.

Acknowledgments. Many readers have sent corrections and suggestions for improvement. Many of the corrections were incorporated into earlier reprintings. Only those that affected pagination were accrued to this new edition. Even though I cannot now remember everyone who wrote, I am grateful to them all. Some sent comments that had significant impact. They were Hande Benson, Eric Denardo, Sudhakar Mandapati, Michael Overton, and Jos Sturm.

Robert J. Vanderbei

Princeton, NJ, USA

Preface to 3rd Edition

It has been almost 7 years since the 2nd edition appeared and the publisher is itching for me to finish a new edition. The previous edition had very few typos. I have fixed them all! Of course, I’ve also added some new material and who knows how many new typos I’ve introduced. The most significant new material is contained in a new chapter on financial applications, which discusses a linear programming variant of the portfolio selection problem and option pricing. I am grateful to Alex d’Aspremont for pointing out that the option pricing problem provides a nice application of duality theory. Finally, I’d like to acknowledge the fact that half (four out of eight) of the typos were reported to me by Trond Steihaug. Thanks Trond!

Robert J. Vanderbei

Princeton, NJ, USA

Preface to 4th Edition

Besides the ongoing tweaking and refining of the language and presentation of the material, this edition also features new material in Chapters 4 and 12 on the average performance of the simplex method.

I’d like to thank Cagin Ararat and Firdevs Ulus for carefully reviewing and commenting on this new material.

Robert J. Vanderbei

Princeton, NJ, USA

The original version of this book was revised.

An erratum to this book can be found at DOI 10.​1007/​978-1-4614-7630-6_​26

Contents

Part 1. Basic Theory: The Simplex Method and Duality 1

Chapter 1.​ Introduction 3

1.​ Managing a Production Facility 3

2.​ The Linear Programming Problem 5

Exercises 7

Notes 9

Chapter 2.​ The Simplex Method 11

1.​ An Example 11

2.​ The Simplex Method 14

3.​ Initialization 16

4.​ Unboundedness 18

5.​ Geometry 19

Exercises 20

Notes 23

Chapter 3.​ Degeneracy 25

1.​ Definition of Degeneracy 25

2.​ Two Examples of Degenerate Problems 25

3.​ The Perturbation/​Lexicographic Method 28

4.​ Bland’s Rule 31

5.​ Fundamental Theorem of Linear Programming 33

6.​ Geometry 33

Exercises 36

Notes 37

Chapter 4.​ Efficiency of the Simplex Method 39

1.​ Performance Measures 39

2.​ Measuring the Size of a Problem 39

3.​ Measuring the Effort to Solve a Problem 40

4.​ Worst-Case Analysis of the Simplex Method 41

5.​ Empirical Average Performance of the Simplex Method 44

Exercises 50

Notes 52

Chapter 5.​ Duality Theory 53

1.​ Motivation:​ Finding Upper Bounds 53

2.​ The Dual Problem 54

3.​ The Weak Duality Theorem 55

4.​ The Strong Duality Theorem 57

5.​ Complementary Slackness 63

6.​ The Dual Simplex Method 64

7.​ A Dual-Based Phase I Algorithm 66

8.​ The Dual of a Problem in General Form 67

9.​ Resource Allocation Problems 69

10.​ Lagrangian Duality 72

Exercises 73

Notes 79

Chapter 6.​ The Simplex Method in Matrix Notation 81

1.​ Matrix Notation 81

2.​ The Primal Simplex Method 83

3.​ An Example 86

4.​ The Dual Simplex Method 90

5.​ Two-Phase Methods 92

6.​ Negative Transpose Property 93

Exercises 95

Notes 97

Chapter 7.​ Sensitivity and Parametric Analyses 99

1.​ Sensitivity Analysis 99

2.​ Parametric Analysis and the Homotopy Method 102

3.​ The Parametric Self-Dual Simplex Method 105

Exercises 107

Notes 109

Chapter 8.​ Implementation Issues 111

1. Solving Systems of Equations: LU-Factorization 111

2.​ Exploiting Sparsity 115

3.​ Reusing a Factorization 119

4.​ Performance Tradeoffs 123

5.​ Updating a Factorization 124

6.​ Shrinking the Bump 127

7.​ Partial Pricing 128

8.​ Steepest Edge 129

Exercises 131

Notes 132

Chapter 9.​ Problems in General Form 133

1.​ The Primal Simplex Method 133

2.​ The Dual Simplex Method 135

Exercises 140

Notes 140

Chapter 10.​ Convex Analysis 141

1.​ Convex Sets 141

2.​ Carathéodory’s Theorem 143

3.​ The Separation Theorem 144

4.​ Farkas’ Lemma 146

5.​ Strict Complementarity 147

Exercises 149

Notes 150

Chapter 11.​ Game Theory 151

1.​ Matrix Games 151

2.​ Optimal Strategies 153

3.​ The Minimax Theorem 155

4.​ Poker 157

Exercises 161

Notes 163

Chapter 12.​ Regression 165

1.​ Measures of Mediocrity 165

2.​ Multidimensional​ Measures:​ Regression Analysis 167

3. L²-Regression 168

4. L¹-Regression 170

5.​ Iteratively Reweighted Least Squares 171

6.​ An Example:​ How Fast Is the Simplex Method?​ 173

Exercises 178

Notes 183

Chapter 13.​ Financial Applications 185

1.​ Portfolio Selection 185

2.​ Option Pricing 190

Exercises 194

Notes 195

Part 2. Network-Type Problems 197

Chapter 14.​ Network Flow Problems 199

1.​ Networks 199

2.​ Spanning Trees and Bases 202

3.​ The Primal Network Simplex Method 206

4.​ The Dual Network Simplex Method 211

5.​ Putting It All Together 214

6.​ The Integrality Theorem 215

Exercises 216

Notes 224

Chapter 15.​ Applications 225

1.​ The Transportation Problem 225

2.​ The Assignment Problem 227

3.​ The Shortest-Path Problem 228

4.​ Upper-Bounded Network Flow Problems 231

5.​ The Maximum-Flow Problem 233

Exercises 235

Notes 239

Chapter 16.​ Structural Optimization 241

1.​ An Example 241

2.​ Incidence Matrices 243

3.​ Stability 244

4.​ Conservation Laws 245

5.​ Minimum-Weight Structural Design 249

6.​ Anchors Away 250

Exercises 252

Notes 254

Part 3. Interior-Point Methods 255

Chapter 17.​ The Central Path 257

Warning:​ Nonstandard Notation Ahead 257

1.​ The Barrier Problem 257

2.​ Lagrange Multipliers 259

3.​ Lagrange Multipliers Applied to the Barrier Problem 262

4.​ Second-Order Information 264

5.​ Existence 264

Exercises 266

Notes 267

Chapter 18.​ A Path-Following Method 269

1.​ Computing Step Directions 269

2.​ Newton’s Method 270

3.​ Estimating an Appropriate Value for the Barrier Parameter 272

4.​ Choosing the Step Length Parameter 272

5.​ Convergence Analysis 274

Exercises 279

Notes 283

Chapter 19.​ The KKT System 285

1.​ The Reduced KKT System 285

2.​ The Normal Equations 286

3.​ Step Direction Decomposition 288

Exercises 290

Notes 291

Chapter 20.​ Implementation Issues for Interior-Point Methods 293

1.​ Factoring Positive Definite Matrices 293

2.​ Quasidefinite Matrices 296

3.​ Problems in General Form 302

Exercises 307

Notes 307

Chapter 21.​ The Affine-Scaling Method 309

1.​ The Steepest Ascent Direction 309

2.​ The Projected Gradient Direction 311

3.​ The Projected Gradient Direction with Scaling 312

4.​ Convergence 316

5.​ Feasibility Direction 317

6.​ Problems in Standard Form 319

Exercises 320

Notes 321

Chapter 22.​ The Homogeneous Self-Dual Method 323

1.​ From Standard Form to Self-Dual Form 323

2.​ Homogeneous Self-Dual Problems 324

3.​ Back to Standard Form 334

4.​ Simplex Method vs.​ Interior-Point Methods 336

Exercises 339

Notes 341

Part 4. Extensions 343

Chapter 23.​ Integer Programming 345

1.​ Scheduling Problems 345

2.​ The Traveling Salesman Problem 346

3.​ Fixed Costs 349

4.​ Nonlinear Objective Functions 350

5.​ Branch-and-Bound 351

Exercises 361

Notes 362

Chapter 24.​ Quadratic Programming 363

1.​ The Markowitz Model 363

2.​ The Dual 367

3.​ Convexity and Complexity 370

4.​ Solution via Interior-Point Methods 373

5.​ Practical Considerations 374

Exercises 376

Notes 378

Chapter 25.​ Convex Programming 379

1.​ Differentiable Functions and Taylor Approximations 379

2.​ Convex and Concave Functions 380

3.​ Problem Formulation 380

4.​ Solution via Interior-Point Methods 381

5.​ Successive Quadratic Approximations 382

6.​ Merit Functions 383

7.​ Parting Words 385

Exercises 385

Notes 388

Erratum E1

Appendix A. Source Listings389

1. The Self-Dual Simplex Method390

2. The Homogeneous Self-Dual Method393

Answers to Selected Exercises395

Bibliography399

Index407

Part 1
Basic Theory: The Simplex Method and Duality

We all love to instruct, though we can teach only what is not worth knowing.— J. Austen

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_1

1. Introduction

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

This book is mostly about a subject called Linear Programming. Before defining what we mean, in general, by a linear programming problem, let us describe a few practical real-world problems that serve to motivate and at least vaguely to define this subject.

1 Managing a Production Facility

Consider a production facility for a manufacturing company. The facility is capable of producing a variety of products that, for simplicity, we enumerate as 1, 2, …, n. These products are constructed/manufactured/produced out of certain raw materials. Suppose that there are m different raw materials, which again we simply enumerate as 1, 2, …, m. The decisions involved in managing/operating this facility are complicated and arise dynamically as market conditions evolve around it. However, to describe a simple, fairly realistic optimization problem, we consider a particular snapshot of the dynamic evolution. At this specific point in time, the facility has, for each raw material i = 1, 2, …, m, a known amount, say b i , on hand. Furthermore, each raw material has at this moment in time a known unit market value. We denote the unit value of the ith raw material by ρ i .

In addition, each product is made from known amounts of the various raw materials. That is, producing one unit of product j requires a certain known amount, say a ij units, of raw material i. Also, the jth final product can be sold at the known prevailing market price of σ j dollars per unit.

Throughout this section we make an important assumption:

The production facility is small relative to the market as a whole and therefore cannot through its actions alter the prevailing market value of its raw materials, nor can it affect the prevailing market price for its products.

We consider two optimization problems related to the efficient operation of this facility (later, in Chapter 5, we will see that these two problems are in fact closely related to each other).

1.1 Production Manager as Optimist

The first problem we wish to consider is the one faced by the company’s production manager. It is the problem of how to use the raw materials on hand. Let us assume that she decides to produce x j units of the jth product, j = 1, 2, …, n. The revenue associated with the production of one unit of product j is σ j . But there is also a cost of raw materials that must be considered. The cost of producing one unit of product j is $$\sum _{i=1}^{m}\rho _{i}a_{ij}$$. Therefore, the net revenue associated with the production of one unit is the difference between the revenue and the cost. Since the net revenue plays an important role in our model, we introduce notation for it by setting

Now, the net revenue corresponding to the production of x j units of product j is simply c j x j , and the total net revenue is

(1.1)

The production planner’s goal is to maximize this quantity. However, there are constraints on the production levels that she can assign. For example, each production quantity x j must be nonnegative, and so she has the constraints

(1.2)

Secondly, she can’t produce more product than she has raw material for. The amount of raw material i consumed by a given production schedule is $$\sum _{j=1}^{n}a_{ij}x_{j}$$, and so she must adhere to the following constraints:

(1.3)

To summarize, the production manager’s job is to determine production values x j , j = 1, 2, …, n, so as to maximize (1.1) subject to the constraints given by (1.2) and (1.3). This optimization problem is an example of a linear programming problem. This particular example is often called the resource allocation problem.

1.2 Comptroller as Pessimist

In another office at the production facility sits an executive called the comptroller. The comptroller’s problem (among others) is to assign a value to the raw materials on hand. These values are needed for accounting and planning purposes to determine the cost of inventory. There are rules about how these values can be set. The most important such rule (and the only one relevant to our discussion) is the following:

The company must be willing to sell the raw materials should an outside firm offer to buy them at a price consistent with these values.

Let w i denote the assigned unit value of the ith raw material, i = 1, 2, …, m. That is, these are the numbers that the comptroller must determine. The lost opportunity cost of having b i units of raw material i on hand is b i w i , and so the total lost opportunity cost is

(1.4)

The comptroller’s goal is to minimize this lost opportunity cost (to make the financial statements look as good as possible). But again, there are constraints. First of all, each assigned unit value w i must be no less than the prevailing unit market value ρ i , since if it were less an outsider would buy the company’s raw material at a price lower than ρ i , contradicting the assumption that ρ i is the prevailing market price. That is,

(1.5)

Similarly,

(1.6)

To see why, suppose that the opposite inequality holds for some specific product j. Then an outsider could buy raw materials from the company, produce product j, and sell it at a lower price than the prevailing market price. This contradicts the assumption that σ j is the prevailing market price, which cannot be lowered by the actions of the company we are studying. Minimizing (1.4) subject to the constraints given by (1.5) and (1.6) is a linear programming problem. It takes on a slightly simpler form if we make a change of variables by letting

In words, y i is the increase in the unit value of raw material i representing the “mark-up” the company would charge should it wish simply to act as a reseller and sell raw materials back to the market. In terms of these variables, the comptroller’s problem is to minimize

subject to

and

Note that we’ve dropped a term, $$\sum _{i=1}^{m}b_{i}\rho _{i}$$, from the objective. It is a constant (the market value of the raw materials), and so, while it affects the value of the function being minimized, it does not have any impact on the actual optimal values of the variables (whose determination is the comptroller’s main interest).

2 The Linear Programming Problem

In the two examples given above, there have been variables whose values are to be decided in some optimal fashion. These variables are referred to as decision variables. They are usually denoted as

In linear programming, the objective is always to maximize or to minimize some linear function of these decision variables

This function is called the objective function. It often seems that real-world problems are most naturally formulated as minimizations (since real-world planners always seem to be pessimists), but when discussing mathematics it is usually nicer to work with maximization problems. Of course, converting from one to the other is trivial both from the modeler’s viewpoint (either minimize cost or maximize profit) and from the analyst’s viewpoint (either maximize ζ or minimize − ζ). Since this book is primarily about the mathematics of linear programming, we usually take the optimist’s view of maximizing the objective function.

In addition to the objective function, the examples also had constraints. Some of these constraints were really simple, such as the requirement that some decision variable be nonnegative. Others were more involved. But in all cases the constraints consisted of either an equality or an inequality associated with some linear combination of the decision variables:

It is easy to convert constraints from one form to another. For example, an inequality constraint

can be converted to an equality constraint by adding a nonnegative variable, w, which we call a slack variable:

On the other hand, an equality constraint

can be converted to inequality form by introducing two inequality constraints:

Hence, in some sense, there is no a priori preference for how one poses the constraints (as long as they are linear, of course). However, we shall also see that, from a mathematical point of view, there is a preferred presentation. It is to pose the inequalities as less-thans and to stipulate that all the decision variables be nonnegative. Hence, the linear programming problem, as we study it, can be formulated as follows:

We refer to linear programs formulated this way as linear programs in standard form. In our aim for consistency, we shall always use m to denote the number of constraints, and n to denote the number of decision variables.

A proposal of specific values for the decision variables is called a solution. A solution $$(x_{1},x_{2},\ldots,x_{n})$$ is called feasible if it satisfies all of the constraints. It is called optimal if in addition it attains the desired maximum. Some problems are just simply infeasible, as the following example illustrates:

Indeed, the second constraint implies that x ₁ + x ₂ ≥ 4. 5, which contradicts the first constraint. If a problem has no feasible solution, then the problem itself is called infeasible.

At the other extreme from infeasible problems, one finds unbounded problems. A problem is unbounded if it has feasible solutions with arbitrarily large objective values. For example, consider

Here, we could set x ₂ to zero and let x ₁ be arbitrarily large. As long as x ₁ is greater than 2 the solution will be feasible, and as it gets large the objective function does too. Hence, the problem is unbounded. In addition to finding optimal solutions to linear programming problems, we shall also be interested in detecting when a problem is infeasible or unbounded.

Exercises

1.1

A steel company must decide how to allocate next week’s time on a rolling mill, which is a machine that takes unfinished slabs of steel as input and can produce either of two semi-finished products: bands and coils. The mill’s two products come off the rolling line at different rates:

Bands

200 tons/h

Coils

140 tons/h.

They also produce different profits:

Bands

$25/ton

Coils

$30/ton.

Based on currently booked orders, the following upper bounds are placed on the amount of each product to produce:

Bands

6,000 tons

Coils

4,000 tons.

Given that there are 40 h of production time available this week, the problem is to decide how many tons of bands and how many tons of coils should be produced to yield the greatest profit. Formulate this problem as a linear programming problem. Can you solve this problem by inspection?

1.2

A small airline, Ivy Air, flies between three cities: Ithaca, Newark, and Boston. They offer several flights but, for this problem, let us focus on the Friday afternoon flight that departs from Ithaca, stops in Newark, and continues to Boston. There are three types of passengers:

(a)

Those traveling from Ithaca to Newark.

(b)

Those traveling from Newark to Boston.

(c)

Those traveling from Ithaca to Boston.

The aircraft is a small commuter plane that seats 30 passengers. The airline offers three fare classes:

(a)

Y class: full coach.

(b)

B class: nonrefundable.

(c)

M class: nonrefundable, 3-week advanced purchase.

Ticket prices, which are largely determined by external influences (i.e., competitors), have been set and advertised as follows:

Based on past experience, demand forecasters at Ivy Air have determined the following upper bounds on the number of potential customers in each of the nine possible origin-destination/fare-class combinations:

The goal is to decide how many tickets from each of the nine origin/ destination/fare-class combinations to sell. The constraints are that the plane cannot be overbooked on either of the two legs of the flight and that the number of tickets made available cannot exceed the forecasted maximum demand. The objective is to maximize the revenue. Formulate this problem as a linear programming problem.

1.3

Suppose that Y is a random variable taking on one of n known values:

Suppose we know that Y either has distribution p given by

or it has distribution q given by

Of course, the numbers p j , j = 1, 2, …, n are nonnegative and sum to one. The same is true for the q j ’s. Based on a single observation of Y, we wish to guess whether it has distribution p or distribution q. That is, for each possible outcome a j , we will assert with probability x j that the distribution is p and with probability 1 − x j that the distribution is q. We wish to determine the probabilities x j , j = 1, 2, …, n, such that the probability of saying the distribution is p when in fact it is q has probability no larger than β, where β is some small positive value (such as 0. 05). Furthermore, given this constraint, we wish to maximize the probability that we say the distribution is p when in fact it is p. Formulate this maximization problem as a linear programming problem.

Notes

The subject of linear programming has its roots in the study of linear inequalities, which can be traced as far back as 1826 to the work of Fourier. Since then, many mathematicians have proved special cases of the most important result in the subject—the duality theorem. The applied side of the subject got its start in 1939 when L.V. Kantorovich noted the practical importance of a certain class of linear programming problems and gave an algorithm for their solution—see Kantorovich (1960). Unfortunately, for several years, Kantorovich’s work was unknown in the West and unnoticed in the East. The subject really took off in 1947 when G.B. Dantzig invented the simplex method for solving the linear programming problems that arose in U.S. Air Force planning problems. The earliest published accounts of Dantzig’s work appeared in 1951 (Dantzig 1951a,b). His monograph (Dantzig 1963) remains an important reference. In the same year that Dantzig invented the simplex method, T.C. Koopmans showed that linear programming provided the appropriate model for the analysis of classical economic theories. In 1975, the Royal Swedish Academy of Sciences awarded the Nobel Prize in economic science to L.V. Kantorovich and T.C. Koopmans “for their contributions to the theory of optimum allocation of resources.” Apparently the academy regarded Dantzig’s work as too mathematical for the prize in economics (and there is no Nobel Prize in mathematics).

The textbooks by Bradley et al. (1977), Bazaraa et al. (1977), and Hillier and Lieberman (1977) are known for their extensive collections of interesting practical applications.

Bibliography

Bazaraa, M., Jarvis, J., and Sherali, H. (1977). Linear programming and network flows (2nd ed.). New York: Wiley.

Bradley, S., Hax, A., and Magnanti, T. (1977). Applied mathematical programming. Reading: Addison Wesley.

Dantzig, G. (1951a). Application of the simplex method to a transportation problem. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 359–373). New York: Wiley.

Dantzig, G. (1951b). A proof of the equivalence of the programming problem and the game problem. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 330–335). New York: Wiley.

Dantzig, G. (1963). Linear programming and extensions. Princeton: Princeton University Press.Crossref

Hillier, F., and Lieberman, G. (1977). Introduction to mathematical programming (2nd ed.). New York: McGraw-Hill.

Kantorovich, L. (1960). Mathematical methods in the organization and planning of production. Management Science, 6, 550–559. Original Russian version appeared in 1939.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_2

2. The Simplex Method

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In this chapter we present the simplex method as it applies to linear programming problems in standard form.

1 An Example

We first illustrate how the simplex method works on a specific example:

(2.1)

We start by adding so-called slack variables. For each of the less-than inequalities in (2.1) we introduce a new variable that represents the difference between the right-hand side and the left-hand side. For example, for the first inequality,

we introduce the slack variable w ₁ defined by

It is clear then that this definition of w ₁, together with the stipulation that w ₁ be nonnegative, is equivalent to the original constraint. We carry out this procedure for each of the less-than constraints to get an equivalent representation of the problem:

(2.2)

Note that we have included a notation, ζ, for the value of the objective function, $$5x_{1} + 4x_{2} + 3x_{3}$$.

The simplex method is an iterative process in which we start with a solution $$x_{1},x_{2},\ldots,w_{3}$$ that satisfies the equations and nonnegativities in (2.2) and then look for a new solution $$\bar{x}_{1},\bar{x}_{2},\ldots,\bar{w}_{3}$$, which is better in the sense that it has a larger objective function value:

We continue this process until we arrive at a solution that can’t be improved. This final solution is then an optimal solution.

To start the iterative process, we need an initial feasible solution $$x_{1},x_{2},\ldots,w_{3}$$. For our example, this is easy. We simply set all the original variables to zero and use the defining equations to determine the slack variables:

The objective function value associated with this solution is ζ = 0.

We now ask whether this solution can be improved. Since the coefficient of x ₁ in the objective function is positive, if we increase the value of x ₁ from zero to some positive value, we will increase ζ. But as we change its value, the values of the slack variables will also change. We must make sure that we don’t let any of them go negative. Since x ₂ and x ₃ are currently set to 0, we see that $$w_{1} = 5 - 2x_{1}$$, and so keeping w ₁ nonnegative imposes the restriction that x ₁ must not exceed 5 ∕ 2. Similarly, the nonnegativity of w ₂ imposes the bound that x ₁ ≤ 11 ∕ 4, and the nonnegativity of w ₃ introduces the bound that x ₁ ≤ 8 ∕ 3. Since all of these conditions must be met, we see that x ₁ cannot be made larger than the smallest of these bounds: x ₁ ≤ 5 ∕ 2. Our new, improved solution then is

This first step was straightforward. It is less obvious how to proceed. What made the first step easy was the fact that we had one group of variables that were initially zero and we had the rest explicitly expressed in terms of these. This property can be arranged even for our new solution. Indeed, we simply must rewrite the equations in (2.2) in such a way that $$x_{1},w_{2},w_{3}$$, and ζ are expressed as functions of w ₁, x ₂, and x ₃. That is, the roles of x ₁ and w ₁ must be swapped. To this end, we use the equation for w ₁ in (2.2) to solve for x ₁:

The equations for w ₂, w ₃, and ζ must also be doctored so that x ₁ does not appear on the right. The easiest way to accomplish this is to do so-called row operations on the equations in (2.2). For example, if we take the equation for w ₂ and subtract two times the equation for w ₁ and then bring the w ₁ term to the right-hand side, we get

Performing analogous row operations for w ₃ and ζ, we can rewrite the equations in (2.2) as

(2.3)

Note that we can recover our current solution by setting the “independent” variables to zero and using the equations to read off the values for the “dependent” variables.

Now we see that increasing w ₁ or x ₂ will bring about a decrease in the objective function value, and so x ₃, being the only variable with a positive coefficient, is the only independent variable that we can increase to obtain a further increase in the objective function. Again, we need to determine how much this variable can be increased without violating the requirement that all the dependent variables remain nonnegative. This time we see that the equation for w ₂ is not affected by changes in x ₃, but the equations for x ₁ and w ₃ do impose bounds, namely x ₃ ≤ 5 and x ₃ ≤ 1, respectively. The latter is the tighter bound, and so the new solution is

The corresponding objective function value is ζ = 13.

Once again, we must determine whether it is possible to increase the objective function further and, if so, how. Therefore, we need to write our equations with ζ, x ₁, w ₂, and x ₃ written as functions of w ₁, x ₂, and w ₃. Solving the last equation in (2.3) for x ₃, we get

Also, performing the appropriate row operations, we can eliminate x ₃ from the other equations. The result of these operations is

(2.4)

We are now ready to begin the third iteration. The first step is to identify an independent variable for which an increase in its value would produce a corresponding increase in ζ. But this time there is no such variable, since all the variables have negative coefficients in the expression for ζ. This fact not only brings the simplex method to a standstill but also proves that the current solution is optimal. The reason is quite simple. Since the equations in (2.4) are completely equivalent to those in (2.2) and, since all the variables must be nonnegative, it follows that ζ ≤ 13 for every feasible solution. Since our current solution attains the value of 13, we see that it is indeed optimal.

1.1 Dictionaries, Bases, Etc.

The systems of equations (2.2), (2.3), and (2.4) that we have encountered along the way are called dictionaries. With the exception of ζ, the variables that appear on the left (i.e., the variables that we have been referring to as the dependent variables) are called basic variables. Those on the right (i.e., the independent variables) are called nonbasic variables. The solutions we have obtained by setting the nonbasic variables to zero are called basic feasible solutions.

2 The Simplex Method

Consider the general linear programming problem presented in standard form:

Our first task is to introduce slack variables and a name for the objective function value:

(2.5)

As we saw in our example, as the simplex method proceeds, the slack variables become intertwined with the original variables, and the whole collection is treated the same. Therefore, it is at times convenient to have a notation in which the slack variables are more or less indistinguishable from the original variables. So we simply add them to the end of the list of x-variables:

That is, we let $$x_{n+i} = w_{i}$$. With this notation, we can rewrite (2.5) as

This is the starting dictionary. As the simplex method progresses, it moves from one dictionary to another in its search for an optimal solution. Each dictionary has m basic variables and n nonbasic variables. Let ℬ denote the collection of indices from $$\{1,2,\ldots,n + m\}$$ corresponding to the basic variables, and let $$\mathcal{N}$$ denote the indices corresponding to the nonbasic variables. Initially, we have $$\mathcal{N} =\{ 1,2,\ldots,n\}$$ and $$\mathcal{B} =\{ n + 1,n + 2,\ldots,n + m\}$$, but this of course changes after the first iteration. Down the road, the current dictionary will look like this:

(2.6)

Note that we have put bars over the coefficients to indicate that they change as the algorithm progresses.

Within each iteration of the simplex method, exactly one variable goes from nonbasic to basic and exactly one variable goes from basic to nonbasic. We saw this process in our example, but let us now describe it in general.

The variable that goes from nonbasic to basic is called the entering variable. It is chosen with the aim of increasing ζ; that is, one whose coefficient is positive: pick k from $$\{j \in \mathcal{N} :\bar{ c}_{j}> 0\}$$. Note that if this set is empty, then the current solution is optimal. If the set consists of more than one element (as is normally the case), then we have a choice of which element to pick. There are several possible selection criteria, some of which will be discussed in the next chapter. For now, suffice it to say that we usually pick an index k having the largest coefficient (which again could leave us with a choice).

The variable that goes from basic to nonbasic is called the leaving variable. It is chosen to preserve nonnegativity of the current basic variables. Once we have decided that x k will be the entering variable, its value will be increased from zero to a positive value. This increase will change the values of the basic variables:

We must ensure that each of these variables remains nonnegative. Hence, we require that

(2.7)

Of these expressions, the only ones that can go negative as x k increases are those for which $$\bar{a}_{ik}$$ is positive; the rest remain fixed or increase. Hence, we can restrict our attention to those i’s for which $$\bar{a}_{ik}$$ is positive. And for such an i, the value of x k at which the expression becomes zero is

Since we don’t want any of these to go negative, we must raise x k only to the smallest of all of these values:

Therefore, with a certain amount of latitude remaining, the rule for selecting the leaving variable is pick l from $$\{i \in \mathcal{B} :\bar{ a}_{ik}> 0\text{ and }\bar{b}_{i}/\bar{a}_{ik}\, is \,minimal\}$$.

The rule just given for selecting a leaving variable describes exactly the process by which we use the rule in practice. That is, we look only at those variables for which $$\bar{a}_{ik}$$ is positive and among those we select one with the smallest value of the ratio $$\bar{b}_{i}/\bar{a}_{ik}$$. There is, however, another, entirely equivalent, way to write this rule which we will often use. To derive this alternate expression we use the convention that $$0/0 = 0$$ and rewrite inequalities (2.7) as

(we shall discuss shortly what happens when one of these ratios is an indeterminate form 0 ∕ 0 as well as what it means if none of the ratios are positive). Since we wish to take the largest possible increase in x k , we see that

Hence, the rule for selecting the leaving variable is as follows: pick l from $$\{i \in \mathcal{B} :\bar{ a}_{ik}/\bar{b}_{i}\, is \,maximal\}$$.

The main difference between these two ways of writing the rule is that in one we minimize the ratio of $$\bar{a}_{ik}$$ to $$\bar{b}_{i}$$ whereas in the other we maximize the reciprocal ratio. Of course, in the minimize formulation one must take care about the sign of the $$\bar{a}_{ik}$$’s. In the remainder of this book we will encounter these types of ratios often. We will always write them in the maximize form since that is shorter to write, acknowledging full well the fact that it is often more convenient, in practice, to do it the other way.

Once the leaving-basic and entering-nonbasic variables have been selected, the move from the current dictionary to the new dictionary involves appropriate row operations to achieve the interchange. This step from one dictionary to the next is called a pivot.

As mentioned above, there is often more than one choice for the entering and the leaving variables. Particular rules that make the choice unambiguous are called pivot rules.

3 Initialization

In the previous section, we presented the simplex method. However, we only considered problems for which the right-hand sides were all nonnegative. This ensured that the initial dictionary was feasible. In this section, we discuss what to do when this is not the case.

Given a linear programming problem

the initial dictionary that we introduced in the preceding section was

The solution associated with this dictionary is obtained by setting each x j to zero and setting each w i equal to the corresponding b i . This solution is feasible if and only if all the right-hand sides are nonnegative. But what if they are not? We handle this difficulty by introducing an auxiliary problem for which

(1)

A feasible dictionary is easy to find and

(2)

The optimal dictionary provides a feasible dictionary for the original problem.

The auxiliary problem is

It is easy to give a feasible solution to this auxiliary problem. Indeed, we simply set x j  = 0, for $$j = 1,\ldots,n$$, and then pick x ₀ sufficiently large. It is also easy to see that the original problem has a feasible solution if and only if the auxiliary problem has a feasible solution with x ₀ = 0. In other words, the original problem has a feasible solution if and only if the optimal solution to the auxiliary problem has objective value zero.

Even though the auxiliary problem clearly has feasible solutions, we have not yet shown that it has an easily obtained feasible dictionary. It is best to illustrate how to obtain a feasible dictionary with an example:

The auxiliary problem is

Next we introduce slack variables and write down an initial infeasible dictionary:

This dictionary is infeasible, but it is easy to convert it into a feasible dictionary. In fact, all we need to do is one pivot with variable x ₀ entering and the “most infeasible variable,” w ₂, leaving the basis:

Note that we now have a feasible dictionary, so we can apply the simplex method as defined earlier in this chapter. For the first step, we pick x ₂ to enter and w ₁ to leave the basis:

Now, for the second step, we pick x ₁ to enter and x ₀ to leave the basis:

This dictionary is optimal for the auxiliary problem. We now drop x ₀ from the equations and reintroduce the original objective function:

Hence, the starting feasible dictionary for the original problem is

As it turns out, this dictionary is optimal for the original problem (since the coefficients of all the variables in the equation for ζ are negative), but we can’t expect to be this lucky in general. All we normally can expect is that the dictionary so obtained will be feasible for the original problem, at which point we continue to apply the simplex method until an optimal solution is reached.

The process of solving the auxiliary problem to find an initial feasible solution is often referred to as Phase I, whereas the process of going from a feasible solution to an optimal solution is called Phase II.

4 Unboundedness

In this section, we discuss how to detect when the objective function value is unbounded.

Let us now take a closer look at the “leaving variable” computation: pick l from $$\{i \in \mathcal{B} :\bar{ a}_{ik}/\bar{b}_{i}\, is\, maximal\}$$. We avoided the issue before, but now we must face what to do if a denominator in one of these ratios vanishes. If the numerator is nonzero, then it is easy to see that the ratio should be interpreted as plus or minus infinity depending on the sign of the numerator. For the case of 0 ∕ 0, the correct convention (as we’ll see momentarily) is to take this as a zero.

What if all of the ratios, $$\bar{a}_{ik}/\bar{b}_{i}$$, are nonpositive? In that case, none of the basic variables will become zero as the entering variable increases. Hence, the entering variable can be increased indefinitely to produce an arbitrarily large objective value. In such situations, we say that the problem is unbounded. For example, consider the following dictionary:

The entering variable is x ₃ and the ratios are

Since none of these ratios is positive, the problem is unbounded.

In the next chapter, we will investigate what happens when some of these ratios take the value + ∞.

5 Geometry

When the number of variables in a linear programming problem is three or less, we can graph the set of feasible solutions together with the level sets of the objective function. From this picture, it is usually a trivial matter to write down the optimal solution. To illustrate, consider the following problem:

Each constraint (including the nonnegativity constraints on the variables) is a half-plane. These half-planes can be determined by first graphing the equation one obtains by replacing the inequality with an equality and then asking whether or not some specific point that doesn’t satisfy the equality (often (0, 0) can be used) satisfies the inequality constraint. The set of feasible solutions is just the intersection of these half-planes. For the problem given above, this set is shown in Figure 2.1. Also shown are two level sets of the objective function. One of them indicates points at which the objective function value is 11. This level set passes through the middle of the set of feasible solutions. As the objective function value increases, the corresponding level set moves to the right. The level set corresponding to the case where the objective function equals 22 is the last level set that touches the set of feasible solutions. Clearly, this is the maximum value of the objective function. The optimal solution is the intersection of this level set with the set of feasible solutions. Hence, we see from Figure 2.1 that the optimal solution is (x ₁, x ₂) = (6, 2).

FIGURE 2.1

The set of feasible solutions together with level sets of the objective function.

Exercises

Solve the following linear programming problems. If you wish, you may check your arithmetic by using the simple online pivot tool:

http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​simple.​html

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

Using today’s date (MMYY) for the seed value, solve 10 initially feasible problems using the online pivot tool:

http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​primal.​html

2.13

Using today’s date (MMYY) for the seed value, solve 10 not necessarily feasible problems using the online pivot tool:

http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​primal_​x0.​html

2.14

Consider the following dictionary:

Using the largest coefficient rule to pick the entering variable, compute the dictionary that results after one pivot.

2.15

Show that the following dictionary cannot be the optimal dictionary for any linear programming problem in which w ₁ and w ₂ are the initial slack variables:

Hint: if it could, what was the original problem from whence it came?

2.16

Graph the feasible region for Exercise 2.8. Indicate on the graph the sequence of basic solutions produced by the simplex method.

2.17

Give an example showing that the variable that becomes basic in one iteration of the simplex method can become nonbasic in the next iteration.

2.18

Show that the variable that becomes nonbasic in one iteration of the simplex method cannot become basic in the next iteration.

2.19

Solve the following linear programming problem:

Here, the numbers p j , j = 1, 2, …, n, are positive and sum to one. The same is true of the q j ’s:

Furthermore (with only minor loss of generality), you may assume that

Finally, the parameter β is a small positive number. See Exercise 1.3 for the motivation for this problem.

Notes

The simplex method was invented by G.B. Dantzig in 1949. His monograph (Dantzig 1963) is the classical reference. Most texts describe the simplex method as a sequence of pivots on a table of numbers called the simplex tableau. Following Chvátal (1983), we have developed the algorithm using the more memorable dictionary notation.

Bibliography

Chvátal, V. (1983). Linear programming. New York: Freeman.

Dantzig, G. (1963). Linear programming and extensions. Princeton: Princeton University Press.Crossref

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_3

3. Degeneracy

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.​1007/​978-1-4614-7630-6_​26

In the previous chapter, we discussed what it means when the ratios computed to calculate the leaving variable are all nonpositive (the problem is unbounded). In this chapter, we take up the more delicate issue of what happens when some of the ratios are infinite (i.e., their denominators vanish).

1 Definition of Degeneracy

We say that a dictionary is degenerate if $$\bar{b}_{i}$$ vanishes for some $$i \in \mathcal{B}$$. A degenerate dictionary could cause difficulties for the simplex method, but it might not. For example, the dictionary we were discussing at the end of the last chapter,

is degenerate, but it was clear that the problem was unbounded and therefore no more pivots were required. Furthermore, had the coefficient of x ₃ in the equation for x ₂ been − 2 instead of 2, then the simplex method would have picked x ₂ for the leaving variable and no difficulties would have been encountered.

Problems arise, however, when a degenerate dictionary produces degenerate pivots. We say that a pivot is a degenerate pivot if one of the ratios in the calculation of the leaving variable is + ∞; i.e., if the numerator is positive and the denominator vanishes. To see what happens, let’s look at a few examples.

2 Two Examples of Degenerate Problems

Here is an example of a degenerate dictionary in which the pivot is also degenerate:

(3.1)

For this dictionary, the entering variable is x ₂ and the ratios computed to determine the leaving variable are 0 and + ∞. Hence, the leaving variable is w ₂, and the fact that the ratio is infinite means that as soon as x ₂ is increased from zero to a positive value, w ₂ will go negative. Therefore, x ₂ can’t really increase. Nonetheless, it can be reclassified from nonbasic to basic (with w ₂ going the other way). Let’s look at the result of this degenerate pivot:

(3.2)

Note that $$\bar{\zeta }$$ remains unchanged at 3. Hence, this degenerate pivot has not produced any increase in the objective function value. Furthermore, the values of the variables haven’t even changed: both before and after this degenerate pivot, they are

But we are now representing this solution in a new way, and perhaps the next pivot will make an improvement, or if not the next pivot perhaps the one after that. Let’s see what happens for the problem at hand. The entering variable for the next iteration is x ₁ and the leaving variable is x ₃, producing a nondegenerate pivot that leads to

These two pivots illustrate what typically happens. When one reaches a degenerate dictionary, it is usual that one or more of the subsequent pivots will be degenerate but that eventually a nondegenerate pivot will lead us away from these degenerate dictionaries. While it is typical for some pivot to “break away” from the degeneracy, the real danger is that the simplex method will make a sequence of degenerate pivots and eventually return to a dictionary that has appeared before, in which case the simplex method enters an infinite loop and never finds an optimal solution. This behavior is called cycling .

Unfortunately, under certain pivoting rules, cycling is possible. In fact, it is possible even when using one of the most popular pivoting rules:

Choose the entering variable as the one with the largest coefficient in the ζ-row of the dictionary.

When two or more variables compete for leaving the basis, pick an x-variable over a slack variable and, if there is a choice, use the variable with the smallest subscript. In other words, reading left to right, pick the first leaving-variable candidate from the list:

However, it is hard to find examples of cycling in which m and n are small. In fact, it has been shown that if a problem has an optimal solution but cycles off-optimum, then the problem must involve dictionaries with at least four (non-slack) variables and two constraints. Here is an example that cycles:

And here is the sequence of dictionaries the above pivot rules produce. For the first pivot, x ₁ enters and w ₁ leaves bringing us to:

For the second iteration, x ₂ enters and w ₂ leaves bringing us to:

For the third iteration, x ₃ enters and x ₁ leaves:

For the fourth iteration, x ₄ enters and x ₂ leaves:

In the fifth iteration, w ₁ enters and x ₃ leaves:

Lastly, for the sixth iteration, w ₂ enters and x ₄ leaves:

Note that we have come back to the original dictionary, and so from here on the simplex method simply cycles through these six dictionaries and never makes any further progress toward an optimal solution. As bad as cycling is, the following theorem tells us that nothing worse can happen:

Theorem 3.1.

If the simplex method fails to terminate, then it must cycle.

Proof.

A dictionary is completely determined by specifying which variables are basic and which are nonbasic. There are only

different possibilities. This number is big, but it is finite. If the simplex method fails to terminate, it must visit some of these dictionaries more than once. Hence, the algorithm cycles.

Note that, if the simplex method cycles, then all the pivots within the cycle must be degenerate. This is easy to see, since the objective function value never decreases. Hence, it follows that all the pivots within the cycle must have the same objective function value, i.e., all of the these pivots must be degenerate.

In practice, degeneracy is common because a zero right-hand side value crops up frequently in real-world problems. Cycling is not as common, but it can happen and therefore must be addressed. Computer implementations of the simplex method in which numbers are represented as integers or as simple rational numbers are at risk of cycling and one of the techniques described in the following sections must be used to avoid the problem. But, most implementations of the simplex method are written with floating point numbers (the computer approximation to a full set of real numbers). With floating point computation there is inevitable round-off error. Hence, a zero appearing as a right-hand side value generally shows up not as an exact zero but rather as a very small number. The result is that the dictionary appears to be slightly off from actually being degenerate and therefore cycling is usually avoided.

3 The Perturbation/Lexicographic Method

As we have seen, there is not just one algorithm called the simplex method. Instead, the simplex method is a whole family of related algorithms from which we can pick a specific instance by specifying what we have been referring to as pivoting rules. We have also seen that, using a very natural pivoting rule, the simplex method can fail to converge to an optimal solution by occasionally cycling indefinitely through a sequence of degenerate pivots associated with a nonoptimal solution.

So this raises a natural question: are there pivoting rules for which the simplex method will, with certainty, either reach an optimal solution or prove that no such solution exists? The answer to this question is yes, and we shall present two choices of such pivoting rules.

The first method is based on the observation that degeneracy is sort of an accident. That is, a dictionary is degenerate if one or more of the $$\bar{b}_{i}$$’s vanish. Our examples have generally used small integers for the data, and in this case it doesn’t seem too surprising that sometimes cancellations occur and we end up with a degenerate dictionary. But each right-hand side could in fact be any real number, and in the world of real numbers the occurrence of any specific number, such as zero, seems to be quite unlikely. So how about perturbing a given problem by adding small random perturbations independently to each of the right-hand sides? If these perturbations are small enough, we can think of them as insignificant and hence not really changing the problem. If they are chosen independently, then the probability of an exact cancellation is zero.

Such random perturbation schemes are used in some implementations, but what we have in mind as we discuss perturbation methods is something a little bit different. Instead of using independent identically distributed random perturbations, let us consider using a fixed perturbation for each constraint, with the perturbation getting much smaller on each succeeding constraint. Indeed, we introduce a small positive number ε ₁ for the first constraint and then a much smaller positive number ε ₂ for the second constraint, etc. We write this as

The idea is that each ε i acts on an entirely different scale from all the other ε i ’s and the data for the problem. What we mean by this is that no linear combination of the ε i ’s using coefficients that might arise in the course of the simplex method can ever produce a number whose size is of the same order as the data in the problem. Similarly, each of the “lower down” ε i ’s can never “escalate” to a higher level. Hence, cancellations can only occur on a given scale. Of course, this complete isolation of scales can never be truly achieved in the real numbers, so instead of actually introducing specific values for the ε i ’s, we simply treat them as abstract symbols having these scale properties.

To illustrate what we mean, let’s look at a specific example. Consider the following degenerate dictionary:

The first step is to introduce symbolic parameters

to get a perturbed problem:

This dictionary is not degenerate. The entering variable is x ₁ and the leaving variable is unambiguously w ₂. The next dictionary is

For the next pivot, the entering variable is x ₂ and, using the fact that ε ₂ ≪ ε ₁, we see that the leaving variable is x ₁. The new dictionary is

This last dictionary is optimal. At this point, we simply drop the symbolic ε i parameters and get an optimal dictionary for the unperturbed problem:

When treating the ε i ’s as symbols, the method is called the lexicographic method. Note that the lexicographic method does not affect the choice of entering variable but does amount to a precise prescription for the choice of leaving variable.

It turns out that the lexicographic method produces a variant of the simplex method that never cycles:

Theorem 3.2.

The simplex method always terminates provided that the leaving variable is selected by the lexicographic rule.

Proof.

It suffices to show that no degenerate dictionary is ever produced. As we’ve discussed before, the ε i ’s operate on different scales and hence can’t cancel with each other. Therefore, we can think of the ε i ’s as a collection of independent variables. Extracting the ε terms from the first dictionary, we see that we start with the following pattern:

And, after several pivots, the ε terms will form a system of linear combinations, say,

Since this system of linear combinations is obtained from the original system by pivot operations and, since pivot operations are reversible, it follows that the rank of the two systems must be the same. Since the original system had rank m, we see that every subsequent system must have rank m. This means that there must be at least one nonzero r ij in every row i, which of course implies that none of the rows can be degenerate. Hence, no dictionary can be degenerate.

4 Bland’s Rule

The second pivoting rule we consider is called Bland’s rule . It stipulates that both the entering and the leaving variable be selected from their respective sets of choices by choosing the variable x k with the smallest index k.

Theorem 3.3.

The simplex method always terminates provided that both the entering and the leaving variable are chosen according to Bland’s rule.

The proof may look rather involved, but the reader who spends the time to understand it will find the underlying elegance most rewarding.

Proof.

It suffices to show that such a variant of the simplex method never cycles. We prove this by assuming that cycling does occur and then showing that this assumption leads to a contradiction. So let’s assume that cycling does occur. Without loss of generality, we may assume that it happens from the beginning. Let D ₀, D ₁, …, D k−1 denote the dictionaries through which the method cycles. That is, the simplex method produces the following sequence of dictionaries:

We say that a variable is fickle if it is in some basis and not in some other basis. Let x t be the fickle variable having the largest index and let D denote a dictionary in D ₀, D ₁, …, D k−1 in which x t leaves the basis. Again, without loss of generality we may assume that D = D ₀. Let x s denote the corresponding entering variable. Suppose that D is recorded as follows:

Since x s is the entering variable and x t is the leaving variable, we have that $$s \in \mathcal{N}$$ and $$t \in \mathcal{B}$$.

Now let D ∗ be a dictionary in D ₁, D ₂, …, D k−1 in which x t enters the basis. Suppose that D ∗ is recorded as follows:

(3.3)

Since all the dictionaries are degenerate, we have that v ∗ = v, and therefore we can write the objective function in (3.3) as

(3.4)

where we have extended the notation c j ∗ to all variables (both original and slack) by setting c j ∗ = 0 for $$j \in \mathcal{B}^{{\ast}}$$.

Ignoring for the moment the possibility that some variables could go negative, consider the solutions obtained by letting x s increase while holding all other variables in $$\mathcal{N}$$ at zero:

The objective function at this point is given by

However, using (3.4), we see that it is also given by

Equating these two expressions for ζ, we see that

Since this equation must be an identity for every y, it follows that the coefficient multiplying y must vanish (as must the right-hand side):

Now, the fact that x s is the entering variable in D implies that

Recall that x t is the fickle variable with the largest index. Since x s is also fickle, we see that s < t. Since x s is not the entering variable in D ∗ (as x t is), we see that

From these last three displayed equations, we get

Hence, there must exist an index $$r \in \mathcal{B}$$ for which

(3.5)

Consequently, c r ∗ ≠ 0 and $$r \in \mathcal{N}^{{\ast}}$$. Hence, x r is fickle and therefore r ≤ t. In fact, r < t, since c t ∗ a ts  > 0. To see that this product is positive, note that both its factors are positive: c t ∗ is positive, since x t is the entering variable in D ∗, and a ts is positive, since x t is the leaving variable in D.

The fact that r < t implies that c r ∗ ≤ 0 (otherwise, according to the smallest index criteria, r would be the entering variable for D ∗). Hence, (3.5) implies that

Now, since each of the dictionaries in the cycle describe the same solution, it follows that every fickle variable is zero in all these dictionaries (since it is clearly zero in a dictionary in which it is nonbasic). In particular, x r  = 0. But in D, x r is basic. Hence,

These last two displayed equations imply that x r was a candidate to be the leaving variable in D, and since r < t, it should have been chosen over x t . This is the contradiction we have been looking for.

5 Fundamental Theorem of Linear Programming

Now that we have a Phase I algorithm and a variant of the simplex method that is guaranteed to terminate, we can summarize the main points of this chapter in the following theorem:

Theorem 3.4.

For an arbitrary linear program in standard form, the following statements are true:

1.

If there is no optimal solution, then the problem is either infeasible or unbounded.

2.

If a feasible solution exists, then a basic feasible solution exists.

3.

If an optimal solution exists, then a basic optimal solution exists.

Proof.

The Phase I algorithm either proves that the problem is infeasible or produces a basic feasible solution. The Phase II algorithm either discovers that the problem is unbounded or finds a basic optimal solution. These statements depend, of course, on applying a variant of the simplex method that does not cycle, which we now know to exist.

6 Geometry

As we saw in the previous chapter, the set of feasible solutions for a problem in two dimensions is the intersection of a number of halfplanes, i.e., a polygon. In three dimensions, the situation is similar. Consider, for example, the following problem:

(3.6)

The set of points satisfying x ₁ + 2x ₃ = 3 is a plane. The inequality x ₁ + 2x ₃ ≤ 3 therefore consists of all points on one side of this plane; that is, it is a halfspace. The same is true for each of the other four inequalities. The feasible set consists of those points in space that satisfy all five inequalities, i.e., those points lying in the intersection of these halfspaces. This set is the polyhedron shown in Figure 3.1. This polyhedron is bordered by five facets , each facet being a portion of one of the planes that was defined by replacing a constraint inequality with an equation. For example, the “front” facet in the figure is a portion of the plane x ₁ + 2x ₃ = 3. The facets acquire a particularly simple description if we introduce slack variables into the problem:

Indeed, each facet corresponds precisely to some variable (either original or slack) vanishing. For instance, the front facet in the figure corresponds to w ₁ = 0 whereas the “left” facet corresponds to x ₂ = 0.

Fig. 3.1

The set of feasible solutions for the problem given by (3.6).

The correspondences can be continued. Indeed, each edge of the polyhedron corresponds to a pair of variables vanishing. For example, the edge lying at the interface of the left and the front facets in the figure corresponds to both w ₁ = 0 and x ₂ = 0.

Going further yet, each vertex of the polyhedron corresponds to three variables vanishing. For instance, the vertex whose coordinates are (1, 0, 1) corresponds to w ₁ = 0, x ₂ = 0, and w ₂ = 0.

Now, let’s think about applying the simplex method to this problem. Every basic feasible solution involves two basic variables and three nonbasic variables. Furthermore, the three nonbasic variables are, by definition, zero in the basic feasible solution. Therefore, for this example, the basic feasible solutions stand in one-to-one correspondence with the vertices of the polyhedron. In fact, applying the simplex method to this problem, one discovers that the sequence of vertices visited by the algorithm is

The example we’ve been considering has the nice property that every vertex is formed by the intersection of exactly three of the facets. But consider now the following problem:

(3.7)

Algebraically, the only difference between this problem and the previous one is that the right-hand side of the first inequality is now a 2 instead of a 3. But look at the polyhedron of feasible solutions shown in Figure 3.2. The vertex (0, 0, 1) is at the intersection of four of the facets, not three as one would “normally” expect. This vertex does not correspond to one basic feasible solution; rather, there are four degenerate basic feasible solutions, each representing it. We’ve seen two of them before. Indeed, dictionaries (3.1) and (3.2) correspond to two of these degenerate dictionaries (in fact, dictionary (3.1) is the dictionary one obtains after one pivot of the simplex method applied to problem (3.7)).

Fig. 3.2

The set of feasible solutions for the (degenerate) problem given by (3.7).

We end by considering the geometric effect of the perturbation method for resolving degeneracy. By perturbing the right-hand sides, one moves the planes that determine the facets. If the moves are random or chosen with vastly different magnitudes (all small), then one would expect that each vertex in the perturbed problem would be determined by exactly three planes. That is, degenerate vertices from the original problem get split into multiple nearby vertices in the perturbed problem. For example, problem (3.6) can be thought of as a perturbation of degenerate problem (3.7) (the perturbation isn’t small, but it also isn’t so large as to obscure the effect). Note how the degenerate vertex in Figure 3.2 appears as two vertices in Figure 3.1.

Exercises

3.1

Solve the following linear program using the perturbation method to resolve degeneracy:

Note: The simple pivot tool with the Lexicographic labels can be used to check your arithmetic:

www.​princeton.​edu/​rvdb/​JAVA/​pivot/​simple.​html

3.2

Solve the following linear program using Bland’s rule to resolve degeneracy:

3.3

Using today’s date (MMYY) for the seed value, solve 10 possibly degenerate problems using the online pivot tool: www.​princeton.​edu/​rvdb/​JAVA/​pivot/​lexico.​html

3.4

Consider the linear programming problems whose right-hand sides are identically zero:

Show that either x j  = 0 for all j is optimal or else the problem is unbounded.

3.5

Consider the following linear program:

Show that this problem has feasible solutions but no vertex solutions. How does this reconcile with the fundamental theorem of linear programming (Theorem 3.4)?

3.6

Suppose that a linear programming problem has the following property: its initial dictionary is not degenerate and, when solved by the simplex method, there is never a tie for the choice of leaving variable.

(a)

Can such a problem have degenerate dictionaries? Explain.

(b)

Can such a problem cycle? Explain.

3.7

Consider the following dictionary:

(a)

List all pairs (x r , x s ) such that x r could be the entering variable and x s could be the leaving variable.

(b)

List all such pairs if the largest-coefficient rule for choosing the entering variable is used.

(c)

List all such pairs if Bland’s rule for choosing the entering and leaving variables is used.

Notes

The first example of cycling was given by Hoffman (1953). The fact that any linear programming problem that cycles must have at least six variables and three constraints was proved by Marshall and Suurballe (1969).

Early proofs of the fundamental theorem of linear programming (Theorem 3.4) were constructive, relying, as in our development, on the existence of a variant of the simplex method that works even in the presense of degeneracy. Hence, finding such variants occupied the attention of early researchers in linear programming. The perturbation method was first suggested by A. Orden and developed independently by Charnes (1952). The essentially equivalent lexicographic method first appeared in Dantzig et al. (1955). Theorem 3.3 was proved by Bland (1977).

For an extensive treatment of degeneracy issues see Gal (1994).

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© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_4

4. Efficiency of the Simplex Method

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In the previous chapter, we saw that the simplex method (with appropriate pivoting rules to guarantee no cycling) will solve any linear programming problem for which an optimal solution exists. In this chapter, we investigate just how fast it will solve a problem of a given size.

1 Performance Measures

Performance measures can be broadly divided into two types:

Worst case

Average case.

As its name implies, a worst-case analysis looks at all problems of a given “size” and asks how much effort is needed to solve the hardest of these problems. Similarly, an average-case analysis looks at the average amount of effort, averaging over all problems of a given size. Worst-case analyses are generally easier than average-case analyses. The reason is that, for worst-case analyses, one simply needs to give an upper bound on how much effort is required and then exhibit a specific example that attains this bound. However, for average-case analyses, one must have a stochastic model of the space of “random linear programming problems” and then be able to say something about the solution effort averaged over all the problems in the sample space. There are two serious difficulties here. The first is that it is not clear at all how one should model the space of random problems. Secondly, given such a model, one must be able to evaluate the amount of effort required to solve every problem in the sample space.

Therefore, worst-case analysis is more tractable than average-case analysis, but it is also less relevant to a person who needs to solve real problems. In this chapter, we will start by giving a detailed worst-case analysis of the simplex method using the largest-coefficient rule to select the entering variable. We will then present and discuss the results of some empirical studies in which millions of linear programming problems were generated randomly and solved by the simplex method. Such studies act as a surrogate for a true average-case analysis.

2 Measuring the Size of a Problem

Before looking at worst cases, we must discuss two issues. First, how do we specify the size of a problem? Two parameters come naturally to mind: m and n. Often, we simply use these two numbers to characterize the size a problem. However, we should mention some drawbacks associated with this choice. First of all, it would be preferable to use only one number to indicate size. Since the data for a problem consist of the constraint coefficients together with the right-hand side and objective function coefficients, perhaps we should use the total number of data elements, which is roughly mn.

The product mn isn’t bad, but what if many or even most of the data elements are zero? Wouldn’t one expect such a problem to be easier to solve? Efficient implementations do indeed take advantage of the presence of lots of zeros, and so an analysis should also account for this. Hence, a good measure might be simply the number of nonzero data elements. This would definitely be an improvement, but one can go further. On a computer, floating-point numbers are all the same size and can be multiplied in the same amount of time. But if a person is to solve a problem by hand (or use unlimited precision computation on a computer), then certainly multiplying 23 by 7 is a lot easier than multiplying 23, 453. 2352 by 86, 833. 245643. So perhaps the best measure of a problem’s size is not the number of data elements, but the actual number of bits needed to store all the data on a computer. This measure is popular among most computer scientists and is usually denoted by L.

However, with a little further abstraction, the size of the data, L, is seen to be ambiguous. As we saw in Chapter 1, real-world problems, while generally large and sparse, usually can be described quite simply and involve only a small amount of true input data that gets greatly expanded when setting the problem up with a constraint matrix, right-hand side, and objective function. So should L represent the number of bits needed to specify the nonzero constraint coefficients, objective coefficients, and right-hand sides, or should it be the number of bits in the original data set plus the number of bits in the description of how this data represents a linear programming problem? No one currently uses this last notion of problem size, but it seems fairly reasonable that they should (or at least that they should seriously consider it). Anyway, our purpose here is merely to mention that these important issues are lurking about, but, as stated above, we shall simply focus on m and n to characterize the size of a problem.

3 Measuring the Effort to Solve a Problem

The second issue to discuss is how one should measure the amount of work required to solve a problem. The best answer is the number of seconds of computer time required to solve the problem, using the computer sitting on one’s desk. Unfortunately, there are (hopefully) many readers of this text, not all of whom use the exact same computer. Even if they did, computer technology changes rapidly, and a few years down the road everyone would be using something entirely different. It would be nice if the National Institute of Standards and Technology (the government organization in charge of setting standards, such as how many threads/inch a standard light bulb should have) would identify a standard computer for the purpose of benchmarking algorithms, but, needless to say, this is not very likely. So the time needed to solve a problem, while the most desirable measure, is not the most practical one here. Fortunately, there is a fairly reasonable substitute. Algorithms are generally iterative processes, and the time to solve a problem can be factored into the number of iterations required to solve the problem times the amount of time required to do each iteration. The first factor, the number of iterations, does not depend on the computer and so is a reasonable surrogate for the actual time. This surrogate is useful when comparing various algorithms within the same general class of algorithms, in which the time per iteration can be expected to be about the same among the algorithms; however, it becomes meaningless when one wishes to compare two entirely different algorithms. For now, we shall measure the amount of effort to solve a linear programming problem by counting the number of iterations, i.e. pivots, needed to solve it.

4 Worst-Case Analysis of the Simplex Method

How bad can the simplex method be in the worst case? Well, we have already seen that for some pivoting rules it can cycle, and hence the worst-case solution time for such variants is infinite. However, what about noncycling variants of the simplex method? Since the simplex method operates by moving from one basic feasible solution to another without ever returning to a previously visited solution, an upper bound on the number of iterations is simply the number of basic feasible solutions, of which there can be at most

For a fixed value of the sum n + m, this expression is maximized when m = n. And how big is it? It is not hard to show that

(see Exercise 4.9). It should be noted that, even though typographically compact, the expression 22n is huge even when n is not very big. For example, for n = 25, we have 2⁵⁰ = 1. 1259 ×10¹⁵.

Our best chance for finding a bad example is to look at the case where m = n. In 1972, V. Klee and G.J. Minty were the first to discover an example in which the simplex method using the largest coefficient rule requires 2 n  − 1 iterations to solve. The example is quite simple to state:

(4.1)

It is instructive to look closely at the constraints. The first three constraints are

The first constraint simply says that x ₁ is no bigger than one. With this in mind, the second constraint says that x ₂ has an upper bound of about 100, depending on how big x ₁ is. Similarly, the third constraint says that x ₃ is roughly no bigger than 10,000 (again, this statement needs some adjustment depending on the sizes of x ₁ and x ₂). Therefore, the constraints are approximately just a set of upper bounds, which means that the feasible region is virtually a stretched n-dimensional hypercube¹:

For this reason, the feasible region for the Klee–Minty problem is often referred to as the Klee–Minty cube. An n-dimensional hypercube has 2 n vertices, and, as we shall see, the simplex method with the largest-coefficient rule will start at one of these vertices and visit every vertex before finally finding the optimal solution.

In order to gain a deeper understanding of the Klee–Minty problem, we first replace the specific right-hand sides, 100i − 1, with more generic values, b i , having the property that

As in the previous chapter, we use the expression a ≪ b to mean that a is so much smaller than b that no factors multiplying a and dividing b that arise in the course of applying the simplex method to the problem at hand can ever make the resulting a as large as the resulting b. Hence, we can think of the b i ’s as independent variables for now (specific values can be chosen later). Next, it is convenient to change each right-hand side replacing b i with

Since the numbers b j , $$j = 1,2,\ldots,i - 1$$ are “small potatoes” compared with b i , this modification to the right-hand sides amounts to a very small perturbation. The right-hand sides still grow by huge amounts as i increases. Finally, we wish to add a constant to the objective function so that our generalized Klee–Minty problem can finally be written as

(4.2)

In Exercise 4.7, you are asked to prove that this problem takes 2 n  − 1 iterations. To start to get a handle on the proof, here are the seven iterations that one gets with n = 3. The initial dictionary is

which is feasible. Using the largest coefficient rule, the entering variable is x ₁. From the fact that each subsequent b i is huge compared with its predecessor it follows that w ₁ is the leaving variable. After the first iteration, the dictionary reads

Now, x ₂ enters and w ₂ leaves, so after the second iteration we get:

After the third iteration

After the fourth iteration

After the fifth iteration

After the sixth iteration

And, finally, after the seventh iteration, we get

which is, of course, optimal.

A few observations should be made. First, every pivot is the swap of an x j with the corresponding w j . Second, every dictionary looks just like the first one with the exception that the w i ’s and the x i ’s have become intertwined and various signs have changed (see Exercise 4.6).

Also note that the final dictionary could have been reached from the initial dictionary in just one pivot if we had selected x ₃ to be the entering variable. But the largest-coefficient rule dictated selecting x ₁. It is natural to wonder whether the largest-coefficient rule could be replaced by some other pivot rule for which the worst-case behavior would be much better than the 2 n behavior of the largest-coefficient rule. So far no one has found such a pivot rule. However, no one has proved that such a rule does not exist either.

Finally, we mention that one desirable property of an algorithm is that it be scale invariant. This means that should the units in which one measures the decision variables in a problem be changed, the algorithm would still behave in exactly the same manner. The simplex method with the largest-coefficient rule is not scale invariant. To see this, consider changing variables in the Klee–Minty problem by putting

In the new variables, the initial dictionary for the n = 3 Klee–Minty problem becomes

Now, the largest-coefficient rule picks variable x ₃ to enter. Variable w ₃ leaves, and the method steps to the optimal solution in just one iteration. There exist pivot rules for the simplex method that are scale invariant. But Klee–Minty-like examples have been found for most proposed alternative pivot rules (whether scale invariant or not). In fact, it is an open question whether there exist pivot rules for which one can prove that no problem instance requires an exponential number of iterations (as a function of m or n).

5 Empirical Average Performance of the Simplex Method

To investigate the empirical average case performance of the simplex method, we generated a large number of random linear programming problems and solved each of them using the simplex method. There are many ways to generate random problems. In this section, we consider one such method and analyze the results.

As we have so dramatically seen, the number of iterations can depend on the specific choice of pivot rule. In the code discussed below, we choose the entering variable to be the one with the largest coefficient. The way we generate random problems makes it unlikely that there will be ties in the choice of largest coefficient (at least after the first pivot). So, even though the program makes a choice, it is not terribly important to articulate what that choice is. Similarly, there is little chance for a tie when choosing a leaving variable, so we do not dwell on this matter either.

We list below the source code. The program is written in a language called matlab—a widely used language whose source code is fairly easy to read, even for those not familiar with the language. So, to get started, here’s how we initialize the data describing a linear programming problem:

m = round(10*exp(log(100)*rand()));

n = round(10*exp(log(100)*rand()));

sigma = 10;

A = round(sigma*(randn(m,n)));

b = round(sigma*abs(randn(m,1)));

c = round(sigma*randn(1,n));

Here, rand() generates a (pseudo) random number uniformly distributed on the interval [0, 1] and round() simply rounds a number to its nearest integer value. The formulas for m and n produce numbers between 10 and 1,000. The formula may seem more complicated than one would expect. For example, one might suggest this simpler formula: m = round(10 + 990*rand()). There is a good reason for the more complicated version. We would like about half of the problems to be between 10 and 100 and the other half to be between 100 and 1,000. Using the simple scheme suggested above, only about 10% of the numbers would be between 10 and 100. The vast majority would be between 100 and 1,000. So, what we want is to have our numbers uniformly distributed when viewed on a logarithmic scale. Our formula for m and n achieves this logarithmically-uniform distribution.

The vector c of objective function coefficients is generated using the function randn(). This function is like rand() but, instead of generating numbers with a uniform distribution, it generates numbers with a Gaussian (aka “normal”) distribution with mean 0 and standard deviation 1. Multiplying such a variable by sigma, which is set to 10, increases the standard deviation to 10. The arguments (1,n) passed to randn() tells the random number generator not to produce just one such number but rather to produce a 1 ×n matrix, i.e. a row vector, of independent instances of these random variables. There is no particular reason to round the coefficients in the row vector c to be integers. The only reason this was done was to make the random problems seem slightly more realistic since many/most real-world problems involve data that is mostly integer-valued. The matrix A and the right-hand side vector b are generated in a manner similar to how c is generated. But, note that the formula for b involves the abs() which returns absolute values so that all elements of b are non-negative. This minor, but important, twist is done to ensure that the starting dictionary is feasible. Here’s the main pivot loop:

iter = 0;

while max(c) > eps,

% pick largest coefficient

[cj, col] = max(c);

Acol = A(:,col);

% select leaving variable

if sum(Acol<-eps) == 0,

opt = -1; % unbounded

’unbounded’

break;

end

nums = b.*(Acol<-eps);

dens = -Acol.*(Acol<-eps);

[t, row] = min(nums./dens);

Arow = A(row,:);

a = A(row,col); % pivot element

A = A - Acol*Arow/a;

A(row,:) = -Arow/a;

A(:,col) = Acol/a;

A(row,col) = 1/a;

brow = b(row);

b = b - brow*Acol/a;

b(row) = -brow/a;

ccol = c(col);

c = c - ccol*Arow/a;

c(col) = ccol/a;

iter = iter+1;

end

In this code, the expression max(c) computes the maximum value of the elements of the vector c. It returns both the maximum value and the index at which this value was attained (the first such index if there are more than one). So, cj is the maximal coefficient and col is the index at which this maximum is attained. Hence col is the entering column. Given the matrix A, the expression A(:,col) denotes the column vector consisting of the elements from the col column of A. Hence, Acol denotes the column of the dictionary associated with the entering variable. The next lines of this short code selects the leaving variable, which is in the row called row. Given the entering column and the leaving row, all that remains is to update the coefficients in the objective function, the right-hand side, and the array of coefficients A. The last few lines encode exactly what one needs to do to carry out a pivot.

The code was run 1,000 times and for each randomly generated problem the values of m, n, and the number of iterations, iter, were saved. Figure 4.1 shows a plot of the number of pivots plotted against the sum m+n. Note that this is a log-log plot. That is, both the horizontal and vertical axes are stretched logarithmically.

FIGURE 4.1

Starting from a primal feasible solution and using the primal simplex method to pivot to an optimal solution, shown here is a log-log plot showing the number of pivots required to reach optimality (or discover that the problem is unbounded) plotted against m + n. Points plotted in blue correspond to problems having an optimal solution whereas points plotted in green correspond to unbounded problems.

The data points shown in blue correspond to problems where an optimal solution was obtained whereas those shown in green correspond to unbounded problems. Clearly, unboundedness is a common occurance for problems generated randomly. In fact, of the 1,000 problems, 501 had optimal solutions and the remaining 499 were unbounded. These numbers suggest that the probability of encountering an unbounded problem is exactly one half. This is likely true but it has not been proven. Further investigation reveals that instances where m > n are almost never unbounded whereas the preponderance of m < n instances are unbounded. Of course, the way m and n were generated, it is true that m < n and m > n are equally likely.

One might be tempted to add code that rounds to zero any right-hand side value that is within a small tolerance of zero. For example, we could add

b = b.*(abs(b)>eps);

just before the line defining row. Doing so forces dictionaries to be degenerate and this can lead to cycling. Experiments show that, with this extra line of code, about 5 out of 1,000 instances will cycle due to degeneracy. Without the extra line of code, no instances cycled.

A second observation is that, for a given value of m + n, there seems to be an effective upper limit on the number of pivots required. Some problems, especially the unbounded ones but also many having an optimal solution, solve in many fewer iterations.

The fact that some problems solved quickly even when m + n was large suggests that perhaps m + n is not the best measure of problem size. Figure 4.2 shows the same data plotted using just n as the measure of problem size. Interestingly, this change dramatically improves the correlation between size and number of iterations for those problems that arrived at an optimal solution. But, the unbounded problems are still spread out quite a bit.

FIGURE 4.2

The same data as shown in Figure 4.1 but plotted against n instead of m + n.

Upon reflection, it seems that perhaps a problem is “easy” if either m or n is small relative to the other. To test this idea, we plot the number of iterations against the minimum of m and n. This plot is shown in Figure 4.3. It appears like we have finally arrived at the best measure of size. Performing a statistical analysis (see Chapter 12), we can empirically derive the straight line through this log-log plot that best matches the data. Separate lines for the blue (optimal) and green (unbounded) points are shown on the graph. For the blue points, the equation is given by

where T denotes the number of pivots required to solve a problem. Taking the exponential of both sides, we get

For the green (unbounded) points, the equation is

In both cases, the rate of growth of T with respect to min(m, n) is “superlinear” as the exponents are both larger than 1. Figure 4.4 is a regular, that is not log-log, plot of the number of simplex pivots versus the minimum of m and n. This plot makes the superlinearity easy to spot.

FIGURE 4.3

The same data as before but plotted against the minimum of m and n.

FIGURE 4.4

The same comparison as in Figure 4.3 but plot linearly rather than log-log. This version makes clear that the number of pivots grows faster than linearly.

Finally, a careful comparison of the blue (optimal solution) data points in Figures 4.2 and 4.3 reveals that they are almost all in exactly the same position in the two plots. The reason is that the vast majority of the problems that had an optimal solution were problems in which n was smaller than m.

Exercises

In solving the following problems, the simple pivot tool can be used to check your arithmetic:

http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​simple.​html

4.1

Compare the performance of the largest-coefficient and the smallest-index pivoting rules on the following linear program:

4.2

Compare the performance of the largest-coefficient and the smallest-index pivoting rules on the following linear program:

4.3

Compare the performance of the largest-coefficient and the smallest-index pivoting rules on the following linear program:

4.4

Solve the Klee–Minty problem (4.1) for n = 3.

4.5

Solve the four variable Klee-Minty problem using the online pivot tool:

http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​kleeminty.​html

4.6

Consider the dictionary

where the b i ’s are as in the Klee–Minty problem (4.2) and where each ε i is ± 1. Fix k and consider the pivot in which x k enters the basis and w k leaves the basis. Show that the resulting dictionary is of the same form as before. How are the new ε i ’s related to the old ε i ’s?

4.7

Use the result of the previous problem to show that the Klee–Minty problem (4.2) requires 2 n  − 1 iterations.

4.8

Consider the Klee–Minty problem (4.2). Suppose that $$b_{i} {=\beta }^{i-1}$$ for some β > 1. Find the greatest lower bound on the set of β’s for which the this problem requires 2 n  − 1 iterations.

4.9

Show that, for any integer n,

4.10

Consider a linear programming problem that has an optimal dictionary in which exactly k of the original slack variables are nonbasic. Show that by ignoring feasibility preservation of intermediate dictionaries this dictionary can be arrived at in exactly k pivots. Don’t forget to allow for the fact that some pivot elements might be zero. Hint: see Exercise 2.15.

4.11

(matlab required.) Modify the matlab code posted at

http://​www.​princeton.​edu/​~rvdb/​LPbook/​complexity/​primalsimplex.​m

so that data elements in A, b, and c are not rounded off to integers. Run the code and compare the results to those shown in Figure 4.3.

4.12

(matlab required.) Modify the matlab code posted at

http://​www.​princeton.​edu/​~rvdb/​LPbook/​complexity/​primalsimplex.​m

so that the output is a log-log plot of the number of pivots versus the product m times n. Run the code and compare the results to those shown in Figure 4.3.

Notes

The first example of a linear programming problem in n variables and n constraints taking 2 n  − 1 iterations to solve was published by Klee and Minty (1972). Several researchers, including Smale (1983), Borgwardt (1982, 1987a), Adler and Megiddo (1985), and Todd (1986), have studied the average number of iterations. For a survey of probabilistic methods, the reader should consult Borgwardt (1987b).

Roughly speaking, a class of problems is said to have polynomial complexity if there is a polynomial p for which every problem of “size” n in the class can be solved by some algorithm in at most p(n) operations. For many years it was unknown whether linear programming had polynomial complexity. The Klee–Minty examples show that, if linear programming is polynomial, then the simplex method is not the algorithm that gives the polynomial bound, since 2 n is not dominated by any polynomial. In 1979, Khachian gave a new algorithm for linear programming, called the ellipsoid method, which is polynomial and therefore established once and for all that linear programming has polynomial complexity. The collection of all problem classes having polynomial complexity is usually denoted by $$\mathcal{P}$$. A class of problems is said to belong to the class $$\mathcal{N}\mathcal{P}$$ if, given a (proposed) solution, one can verify its optimality in a number of operations that is bounded by some polynomial in the “size” of the problem. Clearly, $$\mathcal{P}\subset \mathcal{N}\mathcal{P}$$ (since, if we can solve from scratch in a polynomial amount of time, surely we can verify optimality at least that fast). An important problem in theoretical computer science is to determine whether or not $$\mathcal{P}$$ is a strict subset of $$\mathcal{N}\mathcal{P}$$.

The study of how difficult it is to solve a class of problems is called complexity theory. Readers interested in pursuing this subject further should consult Garey and Johnson (1977).

Bibliography

Adler, I., and Megiddo, N. (1985). A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension. Journal of the ACM, 32, 871–895.Crossref

Borgwardt, K. -H. (1982). The average number of pivot steps required by the simplex-method is polynomial. Zeitschrift für Operations Research, 26, 157–177.

Borgwardt, K. -H. (1987a). Probabilistic analysis of the simplex method. In Operations research proceedings, 16th DGOR meeting, (pp. 564–576).

Borgwardt, K. -H. (1987b). The simplex method—A probabilistic approach. Berlin/Heidelberg/New York: Springer.

Garey, M., and Johnson, D. (1977). Computers and intractability. San Francisco: W.H. Freeman.

Khachian, L. (1979). A polynomial algorithm in linear programming. Doklady Academiia Nauk SSSR, 244, 191–194 (in Russian) (English translation: Soviet Mathematics Doklady, 20, 191–194)

Klee, V., and Minty, G. (1972). How good is the simplex algorithm? In O. Shisha (Ed.), Inequalities–III (pp. 159–175). New York: Academic.

Smale, S. (1983). On the average number of steps of the simplex method of linear programming. Mathematical Programming, 27, 241–262.Crossref

Todd, M. (1986). Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming. Mathematical Programming, 35, 173–192.Crossref

Footnotes

1

More precisely, a hyperrectangle.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_5

5. Duality Theory

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

Associated with every linear program is another called its dual. The dual of this dual linear program is the original linear program (which is then referred to as the primal linear program). Hence, linear programs come in primal/dual pairs. It turns out that every feasible solution for one of these two linear programs gives a bound on the optimal objective function value for the other. These ideas are important and form a subject called duality theory, which is the topic of this chapter.

1 Motivation: Finding Upper Bounds

We begin with an example:

Our first observation is that every feasible solution provides a lower bound on the optimal objective function value, ζ  ∗ . For example, the solution (x ₁, x ₂, x ₃) = (1, 0, 0) tells us that ζ  ∗  ≥ 4. Using the feasible solution (x ₁, x ₂, x ₃) = (0, 0, 3), we see that ζ  ∗  ≥ 9. But how good is this bound? Is it close to the optimal value? To answer, we need to give upper bounds, which we can find as follows. Let’s multiply the first constraint by 2 and add that to 3 times the second constraint:

Now, since each variable is nonnegative, we can compare the sum against the objective function and notice that

Hence, ζ  ∗  ≤ 11. We have localized the search to somewhere between 9 and 11. These bounds leave a gap (within which the optimal solution lies), but they are better than nothing. Furthermore, they can be improved. To get a better upper bound, we again apply the same upper bounding technique, but we replace the specific numbers we used before with variables and then try to find the values of those variables that give us the best upper bound. So we start by multiplying the two constraints by nonnegative numbers, y ₁ and y ₂, respectively. The fact that these numbers are nonnegative implies that they preserve the direction of the inequalities. Hence,

If we stipulate that each of the coefficients of the x i ’s be at least as large as the corresponding coefficient in the objective function,

then we can compare the objective function against this sum (and its bound):

We now have an upper bound, y ₁ + 3y ₂, which we should minimize in our effort to obtain the best possible upper bound. Therefore, we are naturally led to the following optimization problem:

This problem is called the dual linear programming problem associated with the given linear programming problem. In the next section, we will define the dual linear programming problem in general.

2 The Dual Problem

Given a linear programming problem in standard form,

(5.1)

the associated dual linear program is given by

Since we started with (5.1), it is called the primal problem. Our first order of business is to show that taking the dual of the dual returns us to the primal. To see this, we first must write the dual problem in standard form. That is, we must change the minimization into a maximization and we must change the first set of greater-than-or-equal-to constraints into less-than-or-equal-to. Of course, we must effect these changes without altering the problem. To change a minimization into a maximization, we note that to minimize something it is equivalent to maximize its negative and then negate the answer:

To change the direction of the inequalities, we simply multiply through by minus one. The resulting equivalent representation of the dual problem in standard form then is

Now we can take its dual:

which is clearly equivalent to the primal problem as formulated in (5.1).

3 The Weak Duality Theorem

As we saw in our example, the dual problem provides upper bounds for the primal objective function value. This result is true in general and is referred to as the Weak Duality Theorem:

Theorem 5.1.

If $$(x_{1},x_{2},\ldots,x_{n})$$ is feasible for the primal and $$(y_{1},y_{2},\ldots,y_{m})$$ is feasible for the dual, then

Proof.

The proof is a simple chain of obvious inequalities:

where the first inequality follows from the fact that each x j is nonnegative and each c j is no larger than $$\sum _{i}y_{i}a_{ij}$$. The second inequality, of course, holds for similar reasons. □ 

Consider the subset of the real line consisting of all possible values for the primal objective function, and consider the analogous subset associated with the dual problem. The weak duality theorem tells us that the set of primal values lies entirely to the left of the set of dual values. As we shall see shortly, these sets are both closed intervals (perhaps of infinite extent), and the right endpoint of the primal set butts up against the left endpoint of the dual set (see Figure 5.1). That is, there is no gap between the optimal objective function value for the primal and for the dual. The lack of a gap between primal and dual objective values provides a convenient tool for verifying optimality. Indeed, if we can exhibit a feasible primal solution $$(x_{1}^{{\ast}},x_{2}^{{\ast}},\ldots,x_{n}^{{\ast}})$$ and a feasible dual solution $$(y_{1}^{{\ast}},y_{2}^{{\ast}},\ldots,y_{m}^{{\ast}})$$ for which

then we may conclude that each of these solutions is optimal for its respective problem. To see that the primal solution is optimal, consider any other feasible solution $$(x_{1},x_{2},\ldots,x_{n})$$. By the weak duality theorem, we have that

Now, since $$(x_{1}^{{\ast}},x_{2}^{{\ast}},\ldots,x_{n}^{{\ast}})$$ was assumed to be feasible, we see that it must be optimal. An analogous argument shows that the dual solution is also optimal for the dual problem. As an example, consider the solutions x = (0, 0. 25, 3. 25) and y = (1, 3) in our example. Both these solutions are feasible, and both yield an objective value of 10. Hence, the weak duality theorem says that these solutions are optimal.

FIGURE 5.1

The primal objective values are all less than the dual objective values. An important question is whether or not there is a gap between the largest primal value and the smallest dual value.

4 The Strong Duality Theorem

The fact that for linear programming there is never a gap between the primal and the dual optimal objective values is usually referred to as the Strong Duality Theorem:

Theorem 5.2.

If the primal problem has an optimal solution,

then the dual also has an optimal solution,

such that

(5.2)

Carefully written proofs, while attractive for their tightness, sometimes obfuscate the main idea. In such cases, it is better to illustrate the idea with a simple example. Anyone who has taken a course in linear algebra probably already appreciates such a statement. In any case, it is true here as we explain the strong duality theorem.

The main idea that we wish to illustrate here is that, as the simplex method solves the primal problem, it also implicitly solves the dual problem, and it does so in such a way that (5.2) holds.

To see what we mean, let us return to the example discussed in Section 5.1. We start by introducing variables w i , i = 1, 2, for the primal slacks and z j , j = 1, 2, 3, for the dual slacks. Since the inequality constraints in the dual problem are greater-than constraints, each dual slack is defined as a left-hand side minus the corresponding right-hand side. For example,

Therefore, the primal and dual dictionaries are written as follows:

Note that we have recorded the negative of the dual objective function, since we prefer to maximize the objective function appearing in a dictionary. Also note that the numbers in the dual dictionary are simply the negative of the numbers in the primal dictionary arranged with the rows and columns interchanged. Indeed, stripping away everything but the numbers, we have

That is, as a table of numbers, the dual dictionary is the negative transpose of the primal dictionary.

Our goal now is to apply the simplex method to the primal problem and at the same time perform the analogous pivots on the dual problem. We shall discover that the negative-transpose property persists throughout the iterations.

Since the primal dictionary is feasible, no Phase I procedure is necessary. For the first pivot, we pick x ₃ as the entering variable (x ₁ has the largest coefficient, but x ₃ provides the greatest one-step increase in the objective). With this choice, the leaving variable must be w ₂. Since the rows and columns are interchanged in the dual dictionary, we see that “column” x ₃ in the primal dictionary corresponds to “row” z ₃ in the dual dictionary. Similarly, row w ₂ in the primal corresponds to column y ₂ in the dual. Hence, to make an analogous pivot in the dual dictionary, we select y ₂ as the entering variable and z ₃ as the leaving variable. While this choice of entering and leaving variable may seem odd compared to how we have chosen entering and leaving variables before, we should note that our earlier choice was guided by the desire to increase the objective function while preserving feasibility. Here, the dual dictionary is not even feasible, and so such considerations are meaningless. Once we give up those rules for the choice of entering and leaving variables, it is easy to see that a pivot can be performed with any choice of entering and leaving variables provided only that the coefficient on the entering variable in the constraint of the leaving variables does not vanish. Such is the case with the current choice. Hence, we do the pivot in both the primal and the dual. The result is

Note that these two dictionaries still have the property of being negative-transposes of each other. For the next pivot, the entering variable in the primal dictionary is x ₂ (this time there is no choice) and the leaving variable is w ₁. In the dual dictionary, the corresponding entering variable is y ₁ and the leaving variable is z ₂. Doing the pivots, we get

This primal dictionary is optimal, since the coefficients in the objective row are all negative. Looking at the dual dictionary, we see that it is now feasible for the analogous reason. In fact, it is optimal too. Finally, both the primal and dual objective function values are 10.

The situation should now be clear. Given a linear programming problem, which is assumed to possess an optimal solution, first apply the Phase I procedure to get a basic feasible starting dictionary for Phase II. Then apply the simplex method to find an optimal solution. Each primal dictionary generated by the simplex method implicitly defines a corresponding dual dictionary as follows: first write down the negative transpose and then replace each x j with a z j and each w i with a y i . As long as the primal dictionary is not optimal, the implicitly defined dual dictionary will be infeasible. But once an optimal primal dictionary is found, the corresponding dual dictionary will be feasible. Since its objective coefficients are always nonpositive, this feasible dual dictionary is also optimal. Furthermore, at each iteration, the current primal objective function value coincides with the current dual objective function value.

To see why the negative transpose property is preserved from one dictionary to the next, let’s observe the effect of one pivot. To keep notations uncluttered, we consider only four generic entries in a table of coefficients: the pivot element, which we denote by a, one other element on the pivot element’s row, call it b, one other in its column, call it c, and a fourth element, denoted d, chosen to make these four entries into a rectangle. A little thought (and perhaps some staring at the examples above) reveals that a pivot produces the following changes:

The pivot element gets replaced by its reciprocal;

Elements in the pivot row get negated and divided by the pivot element;

Elements in the pivot column get divided by the pivot element; and

All other elements, such as d, get decreased by bc ∕ a.

These effects can be summarized on our generic table as follows:

Now, if we start with a dual dictionary that is the negative transpose of the primal and apply one pivot operation, we get

Note that the resulting dual table is the negative transpose of the resulting primal table. By induction we then conclude that, if we start with this property, it will be preserved throughout the solution process.

Since the strong duality theorem is the most important theorem in this book, we present here a careful proof. Those readers who are satisfied with the above discussion may skip the proof.

Proof of Theorem 5.2.).

It suffices to exhibit a dual feasible solution y  ∗  satisfying (5.2). Suppose we apply the simplex method. We know that the simplex method produces an optimal solution whenever one exists, and we have assumed that one does indeed exist. Hence, the final dictionary will be an optimal dictionary for the primal problem. The objective function in this final dictionary is ordinarily written as

But, since this is the optimal dictionary and we prefer stars to bars for denoting optimal “stuff,” let us write ζ  ∗  instead of $$\bar{\zeta }$$. Also, the collection of nonbasic variables will generally consist of a combination of original variables as well as slack variables. Instead of using $$\bar{c}_{j}$$ for the coefficients of these variables, let us use c j  ∗  for the objective coefficients corresponding to original variables, and let us use d i  ∗  for the objective coefficients corresponding to slack variables. Also, for those original variables that are basic we put c j  ∗  = 0, and for those slack variables that are basic we put d i  ∗  = 0. With these new notations, we can rewrite the objective function as

As we know, ζ  ∗  is the objective function value corresponding to the optimal primal solution:

(5.3)

Now, put

(5.4)

We shall show that $${y}^{{\ast}} = (y_{1}^{{\ast}},y_{2}^{{\ast}},\ldots,y_{m}^{{\ast}})$$ is feasible for the dual problem and satisfies (5.2). To this end, we write the objective function two ways:

Since all these expressions are linear in the variables x j , we can equate the coefficients of each variable appearing on the left-hand side with the corresponding coefficient appearing in the last expression on the right-hand side. We can also equate the constant terms on the two sides. Hence,

(5.5)

(5.6)

Combining (5.3) and (5.5), we get that (5.2) holds. Also, the optimality of the dictionary for the primal problem implies that each c j  ∗  is nonpositive, and hence we see from (5.6) that

By the same reasoning, each d i  ∗  is nonpositive, and so we see from (5.4) that

These last two sets of inequalities are precisely the conditions that guarantee dual feasibility. This completes the proof. □ 

The strong duality theorem tells us that, whenever the primal problem has an optimal solution, the dual problem has one also and there is no duality gap. But what if the primal problem does not have an optimal solution? For example, suppose that it is unbounded. The unboundedness of the primal together with the weak duality theorem tells us immediately that the dual problem must be infeasible. Similarly, if the dual problem is unbounded, then the primal problem must be infeasible. It is natural to hope that these three cases are the only possibilities, because if they were we could then think of the strong duality theorem holding globally. That is, even if, say, the primal is unbounded, the fact that then the dual is infeasible is like saying that the primal and dual have a zero duality gap sitting out at + ∞. Similarly, an infeasible primal together with an unbounded dual could be viewed as a pair in which the gap is zero and sits at − ∞.

But it turns out that there is a fourth possibility that sometimes occurs—it can happen that both the primal and the dual problems are infeasible. For example, consider the following problem:

It is easy to see that both this problem and its dual are infeasible. For these problems, one can think of there being a huge duality gap extending from − ∞ to + ∞.

Duality theory is often useful in that it provides a certificate of optimality. For example, suppose that you were asked to solve a really huge and difficult linear program. After spending weeks or months at the computer, you are finally able to get the simplex method to solve the problem, producing as it does an optimal dual solution y  ∗  in addition to the optimal primal solution x  ∗ . Now, how are you going to convince your boss that your solution is correct? Do you really want to ask her to verify the correctness of your computer programs? The answer is probably not. And in fact it is not necessary. All you need to do is supply the primal and the dual solution, and she only has to check that the primal solution is feasible for the primal problem (that’s easy), the dual solution is feasible for the dual problem (that’s just as easy), and the primal and dual objective values agree (and that’s even easier). Certificates of optimality have also been known to dramatically reduce the amount of time certain underpaid professors have to devote to grading homework assignments!

As we’ve seen, the simplex method applied to a primal problem actually solves both the primal and the dual. Since the dual of the dual is the primal, applying the simplex method to the dual also solves both the primal and the dual problem. Sometimes it is easier to apply the simplex method to the dual, for example, if the dual has an obvious basic feasible solution but the primal does not. We take up this topic in the next chapter.

5 Complementary Slackness

Sometimes it is necessary to recover an optimal dual solution when only an optimal primal solution is known. The following theorem, known as the Complementary Slackness Theorem, can help in this regard.

Theorem 5.3.

Suppose that $$x = (x_{1},x_{2},\ldots,x_{n})$$ is primal feasible and that $$y = (y_{1},y_{2},\ldots,y_{m})$$ is dual feasible. Let $$(w_{1},w_{2},\ldots,w_{m})$$ denote the corresponding primal slack variables, and let $$(z_{1},z_{2},\ldots,z_{n})$$ denote the corresponding dual slack variables. Then x and y are optimal for their respective problems if and only if

(5.7)

Proof.

We begin by revisiting the chain of inequalities used to prove the weak duality theorem:

(5.8)

(5.9)

Recall that the first inequality arises from the fact that each term in the left-hand sum is dominated by the corresponding term in the right-hand sum. Furthermore, this domination is a consequence of the fact that each x j is nonnegative and

Hence, inequality (5.8) will be an equality if and only if, for every $$j = 1,2,\ldots,n$$, either x j  = 0 or $$c_{j} =\sum _{i}y_{i}a_{ij}$$. But since

we see that the alternative to x j  = 0 is simply that z j  = 0. Of course, the statement that at least one of these two numbers vanishes can be succinctly expressed by saying that the product vanishes.

An analogous analysis of inequality (5.9) shows that it is an equality if and only if (5.7) holds. This then completes the proof. □ 

Suppose that we have a nondegenerate primal basic optimal solution

and we wish to find a corresponding optimal solution for the dual. Let

denote the corresponding slack variables, which were probably given along with the x j ’s but if not can be easily obtained from their definition as slack variables:

The dual constraints are

(5.10)

where we have written the inequalities in equality form by introducing slack variables z j , $$j = 1,2,\ldots,n$$. These constraints form n equations in m + n unknowns. But the basic optimal solution (x  ∗ , w  ∗ ) is a collection of n + m variables, many of which are positive. In fact, since the primal solution is assumed to be nondegenerate, it follows that the m basic variables will be strictly positive. The complementary slackness theorem then tells us that the corresponding dual variables must vanish. Hence, of the m + n variables in (5.10), we can set m of them to zero. We are then left with just n equations in n unknowns, which we would expect to have a unique solution that can be solved for. If there is a unique solution, all the components should be nonnegative. If any are negative, this would stand in contradiction to the assumed optimality of x  ∗ .

6 The Dual Simplex Method

In this section, we study what happens if we apply the simplex method to the dual problem. As we saw in our discussion of the strong duality theorem, one can actually apply the simplex method to the dual problem without ever writing down the dual problem or its dictionaries. Instead, the so-called dual simplex method is seen simply as a new way of picking the entering and leaving variables in a sequence of primal dictionaries.

We begin with an example:

The dual of this problem is

Introducing variables w i , i = 1, 2, 3, for the primal slacks and z j , j = 1, 2, for the dual slacks, we can write down the initial primal and dual dictionaries:

As before, we have recorded the negative of the dual objective function, since we prefer to maximize the objective function appearing in a dictionary. More importantly, note that the dual dictionary is feasible, whereas the primal one is not. This suggests that it would be sensible to apply the simplex method to the dual. Let us do so, but as we go we keep track of the analogous pivots applied to the primal dictionary. For example, the entering variable in the initial dual dictionary is y ₂, and the leaving variable then is z ₁. Since w ₂ is complementary to y ₂ and x ₁ is complementary to z ₁, we will use w ₂ and x ₁ as the entering/leaving variables in the primal dictionary. Of course, since w ₂ is basic and x ₁ is nonbasic, w ₂ must be the leaving variable and x ₁ the entering variable—i.e., the reverse of what we have for the complementary variables in the dual dictionary. The result of these pivots is

Continuing to work on the dual, we now see that y ₃ is the entering variable and y ₂ leaves. Hence, for the primal we use w ₃ and w ₂ as the leaving and entering variable, respectively. After pivoting, we have

Now we notice that both dictionaries are optimal.

Of course, in each of the above dictionaries, the table of numbers in each dual dictionary is the negative-transpose of the corresponding primal table. Therefore, we never need to write the dual dictionary; the dual simplex method can be entirely described in terms of the primal dictionaries. Indeed, first we note that the dictionary must be dual feasible. This means that all the coefficients of the nonbasic variables in the primal objective function must be nonpositive. Given this, we proceed as follows. First we select the leaving variable by picking that basic variable whose constant term in the dictionary is the most negative (if there are none, then the current dictionary is optimal). Then we pick the entering variable by scanning across this row of the dictionary and comparing ratios of the coefficients in this row to the corresponding coefficients in the objective row, looking for the largest negated ratio just as we did in the primal simplex method. Once the entering and leaving variable are identified, we pivot to the next dictionary and continue from there. The reader is encouraged to trace the pivots in the above example, paying particular attention to how one determines the entering and leaving variables by looking only at the primal dictionary.

7 A Dual-Based Phase I Algorithm

The dual simplex method described in the previous section provides us with a new Phase I algorithm, which if nothing else is at least more elegant than the one we gave in Chapter 2. Let us illustrate it using an example:

The primal dictionary for this problem is

and even though at this point we realize that we don’t need to look at the dual dictionary, let’s track it anyway:

Clearly, neither the primal nor the dual dictionary is feasible. But by changing the primal objective function, we can easily produce a dual feasible dictionary. For example, let us temporarily change the primal objective function to

Then the corresponding initial dual dictionary is feasible. In fact, it coincides with the dual dictionary we considered in the previous section, so we already know the optimal solution for this modified problem. The optimal primal dictionary is

This primal dictionary is optimal for the modified problem but not for the original problem. However, it is feasible for the original problem, and we can now simply reinstate the intended objective function and continue with Phase II. Indeed,

Hence, the starting dictionary for Phase II is

The entering variable is x ₂. Looking for a leaving variable, we discover that this problem is unbounded. Of course, more typically one would expect to have to do several iterations of Phase II to find the optimal solution (or show unboundedness). Here we just got lucky that the game ended so soon.

It is interesting to note how we detect infeasibility with this new Phase I algorithm. The modified problem is guaranteed always to be dual feasible. It is easy to see that the primal problem is infeasible if and only if the modified problem is dual unbounded (which the dual simplex method will detect just as the primal simplex method detects primal unboundedness).

The two-phase algorithm we have just presented can be thought of as a dual–primal algorithm, since we first apply the dual simplex method to a modified dual feasible problem and then finish off by applying the primal simplex method to the original problem, starting from the feasible dictionary produced by Phase I. One could consider turning this around and doing a primal–dual two-phase algorithm. Here, the right-hand side of the primal problem would be modified to produce an obvious primal feasible solution. The primal simplex method would then be applied. The optimal solution to this primal problem will then be feasible for the original dual problem but will not be optimal for it. But then the dual simplex method can be applied, starting with this dual feasible basis until an optimal solution for the dual problem is obtained.

8 The Dual of a Problem in General Form

In Chapter 1, we saw that linear programming problems can be formulated in a variety of ways. In this section, we derive the form of the dual when the primal problem is not necessarily presented in standard form.

First, let us consider the case where the linear constraints are equalities (and the variables are nonnegative):

(5.11)

As we mentioned in Chapter 1, this problem can be reformulated with inequality constraints by simply writing each equality as two inequalities: one greater-than-or-equal-to and one less-than-or-equal-to:

Then negating each greater-than-or-equal-to constraint, we can put the problem into standard form:

Now that the problem is in standard form, we can write down its dual. Since there are two sets of m inequality constraints, we need two sets of m dual variables. Let’s denote the dual variables associated with the first set of m constraints by y i  + , i = 1, 2, …, m, and the remaining dual variables by y i  − , i = 1, 2, …, m. With these notations, the dual problem is

A moment’s reflection reveals that we can simplify this problem. If we put

the dual problem reduces to

This problem is the dual associated with (5.11). Note what has changed from when we were considering problems in standard form: now the dual variables are not restricted to be nonnegative. And that is the message: equality constraints in the primal yield unconstrained variables (also referred to as free variables) in the dual, whereas inequality constraints in the primal yield nonnegative variables in the dual. Employing the symmetry between the primal and the dual, we can say more: free variables in the primal yield equality constraints in the dual, whereas nonnegative variables in the primal yield inequality constraints in the dual. These rules are summarized in Table 5.1.

Table 5.1

Rules for forming the dual.

Primal

Dual

Equality constraint

Free variable

Inequality constraint

Nonnegative variable

Free variable

Equality constraint

Nonnegative variable

Inequality constraint

9 Resource Allocation Problems

Let us return to the production facility problem studied in Chapter 1. Recall that this problem involves a production facility that can take a variety of raw materials (enumerated i = 1, 2, …, m) and turn them into a variety of final products (enumerated j = 1, 2, …, n). We assume as before that the current market value of a unit of the ith raw material is ρ i , that the current market price for a unit of the jth product is σ j , that producing one unit of product j requires a ij units of raw material i, and that at the current moment in time the facility has on hand b i units of the ith raw material.

The current market values/prices are, by definition, related to each other by the formulas

These equations hold whenever the market is in equilibrium. (Of course, it is crucial to assume here that the collection of “raw materials” appearing on the right-hand side is exhaustive, including such items as depreciation of fixed assets and physical labor.) In the real world, the market is always essentially in equilibrium. Nonetheless, it continually experiences small perturbations that ripple through it and move the equilibrium to new levels.

These perturbations can be from several causes, an important one being innovation. One possible innovation is to improve the production process. This means that the values of some of the a ij ’s are reduced. Now, suddenly there is a windfall profit for each unit of product j produced. This windfall profit is given by

(5.12)

Of course, eventually most producers of these products will take advantage of the same innovation, and once the suppliers get wind of the profits being made, they will get in on the action by raising the price of the raw materials.¹ Nonetheless, there is always a time lag; it is during this time that fortunes are made.

To be concrete, let us assume that the time lag is about 1 month (depending on the industry, this lag time could be considered too short or too long). Suppose also that the production manager decides to produce x j units of product j and that all units produced are sold immediately at their market value. Then the total revenue during this month will be $$\sum _{j}\sigma _{j}x_{j}$$. The value of the raw materials on hand at the beginning of the month was $$\sum _{i}\rho _{i}b_{i}$$. Also, if we denote the new price levels for the raw materials at the end of the month by w i , i = 1, 2, …, m, then the value of any remaining inventory at the end of the month is given by

(if any term is negative, then it represents the cost of purchasing additional raw materials to meet the month’s production requirements—we assume that these additional purchases are made at the new, higher, end-of-month price). The total windfall, call it π, (over all products) for this month can now be written as

(5.13)

Our aim is to choose production levels x j , j = 1, 2, …, n, that maximize this windfall. But our supplier’s aim is to choose prices w i , i = 1, 2, …, m, so as to minimize our windfall. Before studying these optimizations, let us first rewrite the windfall in a more convenient form. As in Chapter 1, let y i denote the increase in the price of raw material i. That is,

(5.14)

Substituting (5.14) into (5.13) and then simplifying notations using (5.12), we see that

(5.15)

To emphasize that π depends on each of the x j ’s and on the y i ’s, we sometimes write it as π(x ₁, …, x n , y ₁, …, y m ).

Now let us return to the competing optimizations. Given x j for j = 1, 2, …, n, the suppliers react to minimize π(x ₁, …, x n , y ₁, …, y m ). Looking at (5.15), we see that for any resource i in short supply, that is,

the suppliers will jack up the price immensely (i.e., y i  = ∞). To avoid this obviously bad situation, the production manager will be sure to set the production levels so that

On the other hand, for any resource i that is not exhausted during the windfall month, that is,

the suppliers will have no incentive to change the prevailing market price (i.e., y i  = 0). Therefore, from the production manager’s point of view, the problem reduces to one of maximizing

subject to the constraints that

This is just our usual primal linear programming problem. This is the problem that the production manager needs to solve in anticipation of adversarial suppliers.

Now let us look at the problem from the suppliers’ point of view. Rearranging the terms in (5.15) by writing

(5.16)

we see that if the suppliers set prices in such a manner that a windfall remains on the jth product even after the price adjustment, that is,

then the production manager would be able to generate for the facility an arbitrarily large windfall by producing a huge amount of the jth product (i.e., x j  = ∞). We assume that this is unacceptable to the suppliers, and so they will determine their price increases so that

Also, if the suppliers set the price increases too high so that the production facility will lose money by producing product j, that is,

then the production manager would simply decide not to engage in that activity. That is, she would set x j  = 0. Hence, the first term in (5.16) will always be zero, and so the optimization problem faced by the suppliers is to minimize

subject to the constraints that

This is precisely the dual of the production manager’s problem!

As we’ve seen earlier with the strong duality theorem, if the production manager’s problem has an optimal solution, then so does the suppliers’ problem, and the two objectives agree. This means than an equilibrium can be reestablished by setting the production levels and the price hikes according to the optimal solutions to these two linear programming problems.

10 Lagrangian Duality

The analysis of the preceding section is an example of a general technique that forms the foundation of a subject called Lagrangian duality, which we shall briefly describe.

Let us start by summarizing what we did. It was quite simple. The analysis revolved around a function

To streamline notations, let x stand for the entire collection of variables x ₁, x ₂, …, x n and let y stand for the collection of y i ’s so that we can write π(x, y) in place of π(x ₁, …, x n , y ₁, …, y m ). Written with these notations, we showed in the previous section that

We also showed that the inner optimization could in both cases be solved explicitly, that the max–min problem reduced to a linear programming problem, and that the min–max problem reduced to the dual linear programming problem.

One could imagine trying to carry out the same program for functions π that don’t necessarily have the form shown above. In the general case, one needs to consider each step carefully. The max–min problem is called the primal problem, and the min–max problem is called the dual problem. However, it may or may not be true that these two problems have the same optimal objective values. In fact, the subject is interesting because one can indeed state specific, verifyable conditions for which the two problems do agree. Also, one would like to be able to solve the inner optimizations explicitly so that the primal problem can be stated as a pure maximization problem and the dual can be stated as a pure minimization problem. This, too, is often doable. There are various ways in which one can extend the notions of duality beyond the context of linear programming. The one just described is referred to as Lagrangian duality. It is perhaps the most important such extension.

Exercises

In solving the following problems, the advanced pivot tool can be used to check your arithmetic:

http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​advanced.​html

5.1

What is the dual of the following linear programming problem:

5.2

Illustrate Theorem 5.2 on the problem in Exercise 2.9.

5.3

Illustrate Theorem 5.2 on the problem in Exercise 2.1.

5.4

Illustrate Theorem 5.2 on the problem in Exercise 2.2.

5.5

Consider the following linear programming problem:

Suppose that, in solving this problem, you have arrived at the following dictionary:

(a)

Write down the dual problem.

(b)

In the dictionary shown above, which variables are basic? Which are nonbasic?

(c)

Write down the primal solution corresponding to the given dictionary. Is it feasible? Is it degenerate?

(d)

Write down the corresponding dual dictionary.

(e)

Write down the dual solution. Is it feasible?

(f)

Do the primal/dual solutions you wrote above satisfy the complementary slackness property?

(g)

Is the current primal solution optimal?

(h)

For the next (primal) pivot, which variable will enter if the largest coefficient rule is used? Which will leave? Will the pivot be degenerate?

5.6

Solve the following linear program:

5.7

Solve the linear program given in Exercise 2.3 using the dual–primal two-phase algorithm.

5.8

Solve the linear program given in Exercise 2.4 using the dual–primal two-phase algorithm.

5.9

Solve the linear program given in Exercise 2.6 using the dual–primal two-phase algorithm.

5.10

Using today’s date (MMYY) for the seed value, solve 10 problems using the dual phase I primal phase II simplex method:

http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​dp2phase.​html

5.11

Using today’s date (MMYY) for the seed value, solve 10 problems using the primal phase I dual phase II simplex method:

http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​pd2phase.​html

5.12

For x and y in $$\mathbb{R}$$, compute

and note whether or not they are equal.

5.13

Consider the following process. Starting with a linear programming problem in standard form,

first form its dual:

Then replace the minimization in the dual with a maximization to get a new linear programming problem, which we can write in standard form as follows:

If we identify a linear programming problem with its data, (a ij , b i , c j ), the above process can be thought of as a transformation T on the space of data defined by

Let ζ  ∗ (a ij , b i , c j ) denote the optimal objective function value of the standard-form linear programming problem having data (a ij , b i , c j ). By strong duality together with the fact that a maximization dominates a minimization, it follows that

Now if we repeat this process, we get

and hence that

But the first and the last entry in this chain of inequalities are equal. Therefore, all these inequalities would seem to be equalities. While this outcome could happen sometimes, it certainly isn’t always true. What is the error in this logic? Can you state a (correct) nontrivial theorem that follows from this line of reasoning? Can you give an example where the four inequalities are indeed all equalities?

5.14

Consider the following variant of the resource allocation problem:

(5.17)

As usual, the c j ’s denote the unit prices for the products and the b i ’s denote the number of units on hand for each raw material. In this variant, the u j ’s denote upper bounds on the number of units of each product that can be sold at the set price. Now, let’s assume that the raw materials have not been purchased yet and it is part of the problem to determine the b i ’s. Let p i , i = 1, 2, …, m denote the price for raw material i. The problem then becomes an optimization over both the x j ’s and the b i ’s:

(a)

Show that this problem always has an optimal solution.

(b)

Let y i  ∗ (b), i = 1, 2, …, m, denote optimal dual variables for the original resource allocation problem (5.17). Note that we’ve explicitly indicated that these dual variables depend on the b’s. Also, we assume that problem (5.17) is both primal and dual non-degenerate so the y i  ∗ (b) is uniquely defined. Show that the optimal value of the b i ’s, call them b i  ∗ ’s, satisfy

Hint: You will need to use the fact that, for resource allocation problems, we have a ij ≥ 0 for all i, and all j.

5.15

Consider the following linear program:

Here, the numbers p j , j = 1, 2, …, n are positive and sum to one. The same is true of the q j ’s:

Furthermore, assume that

and that the parameter β is a small positive number. Let $$k =\min \{ j : q_{j+1} + \cdots + q_{n} \leq \beta \}$$. Let y ₀ denote the dual variable associated with the constraint involving β, and let y j denote the dual variable associated with the upper bound of 1 on variable x j . Using duality theory, show that the optimal values of the primal and dual variables are given by

See Exercise 1.3 for the motivation for this problem.

5.16

Diet Problem. An MIT graduate student was trying to make ends meet on a very small stipend. He went to the library and looked up the National Research Council’s publication entitled “Recommended Dietary Allowances” and was able to determine a minimum daily intake quantity of each essential nutrient for a male in his weight and age category. Let m denote the number of nutrients that he identified as important to his diet, and let b i for i = 1, 2, …, m denote his personal minimum daily requirements. Next, he made a list of his favorite foods (which, except for pizza and due mostly to laziness and ineptitude in the kitchen, consisted almost entirely of frozen prepared meals). He then went to the local grocery store and made a list of the unit price for each of his favorite foods. Let us denote these prices as c j for j = 1, 2, …, n. In addition to prices, he also looked at the labels and collected information about how much of the critical nutrients are contained in one serving of each food. Let us denote by a ij the amount of nutrient i contained in food j. (Fortunately, he was able to call his favorite pizza delivery service and get similar information from them.) In terms of this information, he formulated the following linear programming problem:

Formulate the dual to this linear program. Can you introduce another person into the above story whose problem would naturally be to solve the dual?

5.17

Saddle points. A function h(y) defined for $$y \in \mathbb{R}$$ is called strongly convex if

h′(y) > 0 for all y ∈ ℝ,

$$\lim _{y\rightarrow -\infty }h^{\prime}(y) = -\infty $$, and

$$\lim _{y\rightarrow \infty }h^{\prime}(y) = \infty $$.

A function h is called strongly concave if − h is strongly convex. Let π(x, y), be a function defined for $$(x,y) \in {\mathbb{R}}^{2}$$ and having the following form

where f is strongly concave and g is strongly convex. Using elementary calculus

1.

Show that there is one and only one point $$({x}^{{\ast}},{y}^{{\ast}}) \in {\mathbb{R}}^{2}$$ at which the gradient of π,

vanishes. Hint: From the two equations obtained by setting the derivatives to zero, derive two other relations having the form x = ϕ(x) and y = ψ(y). Then study the functions ϕ and ψ to show that there is one and only one solution.

2.

Show that

where (x  ∗ , y  ∗ ) denotes the “critical point” identified in Part I above. (Note: Be sure to check the signs of the second derivatives for both the inner and the outer optimizations.)

Associated with each strongly convex function h is another function, called the Legendre transform of h and denoted by L h , defined by

3.

Using elementary calculus, show that L h is strongly convex.

4.

Show that

and that

5.

Show that the Legendre transform of the Legendre transform of a function is the function itself. That is,

Hint: This can be proved from scratch but it is easier to use the result of Part 2 above.

Notes

The idea behind the strong duality theorem can be traced back to conversations between G.B. Dantzig and J. von Neumann in the fall of 1947, but an explicit statement did not surface until the paper of Gale et al. (1951). The term primal problem was coined by G.B. Dantzig’s father, T. Dantzig. The dual simplex method was first proposed by Lemke (1954).

The solution to Exercise 5.13 (which is left to the reader to supply) suggests that a random linear programming problem is infeasible with probability 1/4, unbounded with probability 1/4, and has an optimal solution with probability 1/2.

Bibliography

Gale, D., Kuhn, H., and Tucker, A. (1951). Linear programming and the theory of games. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 317–329). New York: Wiley.

Lemke, C. (1954). The dual method of solving the linear programming problem. Naval Research Logistics Quarterly, 1, 36–47.Crossref

Footnotes

1

One could take the prices of raw materials as fixed and argue that the value of the final products will fall. It doesn’t really matter which view one adopts, since prices are relative anyway. The point is simply that the difference between the price of the raw materials and the price of the final products must narrow due to this innovation.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_6

6. The Simplex Method in Matrix Notation

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

So far, we have avoided using matrix notation to present linear programming problems and the simplex method. In this chapter, we shall recast everything into matrix notation. At the same time, we will emphasize the close relations between the primal and the dual problems.

1 Matrix Notation

As usual, we begin our discussion with the standard-form linear programming problem:

In the past, we have generally denoted slack variables by w i ’s but have noted that sometimes it is convenient just to string them onto the end of the list of original variables. Such is the case now, and so we introduce slack variables as follows:

With these slack variables, we now write our problem in matrix form:

where

(6.1)

(6.2)

As we know, the simplex method is an iterative procedure in which each iteration is characterized by specifying which m of the n + m variables are basic. As before, we denote by $$\mathcal{B}$$ the set of indices corresponding to the basic variables, and we denote by $$\mathcal{N}$$ the remaining nonbasic indices.

In component notation, the ith component of Ax can be broken up into a basic part and a nonbasic part:

(6.3)

We wish to introduce a notation for matrices that will allow us to break up the matrix product Ax analogously. To this end, let B denote an m ×m matrix whose columns consist precisely of the m columns of A that are associated with the basic variables. Similarly, let N denote an m ×n matrix whose columns are the n nonbasic columns of A. Then we write A in a partitioned-matrix form as follows:

Strictly speaking, the matrix on the right does not equal the A matrix. Instead, it is the A matrix with its columns rearranged in such a manner that all the columns associated with basic variables are listed first followed by the nonbasic columns. Nonetheless, as long as we are consistent and rearrange the rows of x in the same way, then no harm is done. Indeed, let us similarly rearrange the rows of x and write

Then the following separation of Ax into a sum of two terms is true and captures the same separation into basic and nonbasic parts as we had in (6.3):

By similarly partitioning c, we can write

2 The Primal Simplex Method

A dictionary has the property that the basic variables are written as functions of the nonbasic variables. In matrix notation, we see that the constraint equations

can be written as

The fact that the basic variables $$x_{\mathcal{B}}$$ can be written as a function of the nonbasic variables $$x_{\mathcal{N}}$$ is equivalent to the fact that the matrix B is invertible, and hence,

(6.4)

(The fact that B is invertible means that its m column vectors are linearly independent and therefore form a basis for $${\mathbb{R}}^{m}$$—this is why the basic variables are called basic, in case you were wondering.) Similarly, the objective function can be written as

(6.5)

Combining (6.5) and (6.4), we see that we can write the dictionary associated with basis $$\mathcal{B}$$ as

(6.6)

Comparing against the component-form notation of Chapter 2 (see (2.​6)), we make the following identifications:

where the bracketed expressions on the right denote vectors and matrices with the index i running over $$\mathcal{B}$$ and the index j running over $$\mathcal{N}$$. The basic solution associated with dictionary (6.6) is obtained by setting $$x_{\mathcal{N}}$$ equal to zero:

(6.7)

As we saw in the last chapter, associated with each primal dictionary there is a dual dictionary that is simply the negative-transpose of the primal. However, to have the negative-transpose property, it is important to correctly associate complementary pairs of variables. So first we recall that, for the current discussion, we have appended the primal slack variables to the end of the original variables:

Also recall that the dual slack variables are complementary to the original primal variables and that the original dual variables are complementary to the primal slack variables. Therefore, to maintain the desired complementarity condition between like indices in the primal and the dual, we need to relabel the dual variables and append them to the end of the dual slacks:

With this relabeling of the dual variables, the dual dictionary corresponding to (6.6) is

The dual solution associated with this dictionary is obtained by setting $$z_{\mathcal{B}}$$ equal to zero:

(6.8)

Using (6.7) and (6.8) and introducing the shorthand

(6.9)

we see that we can write the primal dictionary succinctly as

(6.10)

The associated dual dictionary then has a very symmetric appearance:

(6.11)

The (primal) simplex method can be described briefly as follows. The starting assumptions are that we are given

1.

A partition of the n + m indices into a collection $$\mathcal{B}$$ of m basic indices and a collection $$\mathcal{N}$$ of n nonbasic ones with the property that the basis matrix B is invertible,

2.

An associated current primal solution $$x_{\mathcal{B}}^{{\ast}}\geq 0$$ (and $$x_{\mathcal{N}}^{{\ast}} = 0$$), and

3.

An associated current dual solution $$z_{\mathcal{N}}^{{\ast}}$$ (with $$z_{\mathcal{B}}^{{\ast}} = 0$$)

such that the dictionary given by (6.10) represents the primal objective function and the primal constraints. The simplex method then produces a sequence of steps to “adjacent” bases such that the current value ζ  ∗  of the objective function ζ increases at each step (or, at least, would increase if the step size were positive), updating $$x_{\mathcal{B}}^{{\ast}}$$ and $$z_{\mathcal{N}}^{{\ast}}$$ along the way. Two bases are said to be adjacent to each other if they differ in only one index. That is, given a basis $$\mathcal{B}$$, an adjacent basis is determined by removing one basic index and replacing it with a nonbasic index. The index that gets removed corresponds to the leaving variable, whereas the index that gets added corresponds to the entering variable.

One step of the simplex method is called an iteration. We now elaborate further on the details by describing one iteration as a sequence of specific steps.

Step 1. Check for Optimality. If $$z_{\mathcal{N}}^{{\ast}}\geq 0$$, stop. The current solution is optimal. To see this, first note that the simplex method always maintains primal feasibility and complementarity. Indeed, the primal solution is feasible, since $$x_{\mathcal{B}}^{{\ast}}\geq 0$$ and $$x_{\mathcal{N}} = 0$$ and the dictionary embodies the primal constraints. Also, the fact that $$x_{\mathcal{N}}^{{\ast}} = 0$$ and $$z_{\mathcal{B}}^{{\ast}} = 0$$ implies. Hence, all that is required for optimality is dual feasibility. But by looking at the associated dual dictionary (6.11), we see that the dual solution is feasible if and only if $$z_{\mathcal{N}}^{{\ast}}\geq 0$$.

Step 2. Select Entering Variable. Pick an index $$j \in \mathcal{N}$$ for which z j  ∗  < 0. Variable x j is the entering variable.

Step 3. Compute Primal Step Direction $$\Delta x_{\mathcal{B}}$$. Having selected the entering variable, it is our intention to let its value increase from zero. Hence, we let

where we follow the common convention of letting e j denote the unit vector that is zero in every component except for a one in the position associated with index j (note that, because of our index rearrangement conventions, this is not generally the jth element of the vector). Then from (6.10), we have that

Hence, we see that the step direction $$\Delta x_{\mathcal{B}}$$ for the primal basic variables is given by

Step 4. Compute Primal Step Length. We wish to pick the largest t ≥ 0 for which every component of $$x_{\mathcal{B}}$$ remains nonnegative. That is, we wish to pick the largest t for which

Since, for each $$i \in {\mathcal{B}}^{{\ast}}$$, x i  ∗  ≥ 0 and t ≥ 0, we can divide both sides of the above inequality by these numbers and preserve the sense of the inequality. Therefore, doing this division, we get the requirement that

We want to let t be as large as possible, and so 1 ∕ t should be made as small as possible. The smallest possible value for 1 ∕ t that satisfies all the required inequalities is obviously

Hence, the largest t for which all of the inequalities hold is given by

As always, the correct convention for 0 ∕ 0 is to set such ratios to zero. Also, if the maximum is less than or equal to zero, we can stop here—the primal is unbounded.

Step 5. Select Leaving Variable. The leaving variable is chosen as any variable x i , $$i \in \mathcal{B}$$, for which the maximum in the calculation of t is obtained.

Step 6. Compute Dual Step Direction $$\Delta z_{\mathcal{N}}$$. Essentially all that remains is to explain how $$z_{\mathcal{N}}^{{\ast}}$$ changes. To see how, it is convenient to look at the dual dictionary. Since in that dictionary z i is the entering variable, we see that

Step 7. Compute Dual Step Length. Since we know that z j is the leaving variable in the dual dictionary, we see immediately that the step length for the dual variables is

Step 8. Update Current Primal and Dual Solutions. We now have everything we need to update the data in the dictionary:

and

Step 9. Update Basis. Finally, we update the basis:

We close this section with the important remark that the simplex method as presented here, while it may look different from the component-form presentation given in Chapter 2, is in fact mathematically identical to it. That is, given the same set of pivoting rules and starting from the same primal dictionary, the two algorithms will generate exactly the same sequence of dictionaries.

3 An Example

In case the reader is feeling at this point that there are too many letters and not enough numbers, here is an example that illustrates the matrix approach to the simplex method. The problem we wish to solve is

The matrix A is given by

(Note that some zeros have not been shown.) The initial sets of basic and nonbasic indices are

Corresponding to these sets, we have the submatrices of A:

From (6.7) we see that the initial values of the basic variables are given by

and from (6.8) the initial nonbasic dual variables are simply

Since $$x_{\mathcal{B}}^{{\ast}}\geq 0$$, the initial solution is primal feasible, and hence we can apply the simplex method without needing any Phase I procedure.

3.1 First Iteration

Step 1. Since $$z_{\mathcal{N}}^{{\ast}}$$ has some negative components, the current solution is not optimal.

Step 2. Since $$z_{1}^{{\ast}} = -4$$ and this is the most negative of the two nonbasic dual variables, we see that the entering index is

Step 3.

Step 4.

Step 5. Since the ratio that achieved the maximum in Step 4 was the first ratio and this ratio corresponds to basis index 3, we see that

Step 6.

Step 7.

Step 8.

Step 9. The new sets of basic and nonbasic indices are

Corresponding to these sets, we have the new basic and nonbasic submatrices of A,

and the new basic primal variables and nonbasic dual variables:

3.2 Second Iteration

Step 1. Since $$z_{\mathcal{N}}^{{\ast}}$$ has some negative components, the current solution is not optimal.

Step 2. Since $$z_{2}^{{\ast}} = -7$$, we see that the entering index is

Step 3.

Step 4.

Step 5. Since the ratio that achieved the maximum in Step 4 was the second ratio and this ratio corresponds to basis index 4, we see that

Step 6.

Step 7.

Step 8.

Step 9. The new sets of basic and nonbasic indices are

Corresponding to these sets, we have the new basic and nonbasic submatrices of A,

and the new basic primal variables and nonbasic dual variables:

3.3 Third Iteration

Step 1. Since $$z_{\mathcal{N}}^{{\ast}}$$ has some negative components, the current solution is not optimal.

Step 2. Since $$z_{3}^{{\ast}} = -10$$, we see that the entering index is

Step 3.

Step 4.

Step 5. Since the ratio that achieved the maximum in Step 4 was the third ratio and this ratio corresponds to basis index 5, we see that

Step 6.

Step 7.

Step 8.

Step 9. The new sets of basic and nonbasic indices are

Corresponding to these sets, we have the new basic and nonbasic submatrices of A,

and the new basic primal variables and nonbasic dual variables:

3.4 Fourth Iteration

Step 1. Since $$z_{\mathcal{N}}^{{\ast}}$$ has all nonnegative components, the current solution is optimal. The optimal objective function value is

It is undoubtedly clear at this point that the matrix approach, as we have presented it, is quite a bit more tedious than the dictionary manipulations with which we are quite familiar. The reason is that, with the dictionary approach, dictionary entries get updated from one iteration to the next and the updating process is fairly easy, whereas with the matrix approach, we continually compute everything from scratch and therefore end up solving many systems of equations. In the next chapter, we will deal with this issue and show that these systems of equations don’t really have to be solved from scratch each time; instead, there is a certain updating that can be done that is quite analogous to the updating of a dictionary. However, before we take up such practical considerations, let us finish our general discussion of the simplex method by casting the dual simplex method into matrix notation and discussing some related issues.

4 The Dual Simplex Method

In the presentation of the primal simplex method given in the previous section, we tried to make the symmetry between the primal and the dual problems as evident as possible. One advantage of this approach is that we can now easily write down the dual simplex method. Instead of assuming that the primal dictionary is feasible ($$x_{\mathcal{B}}^{{\ast}}\geq 0$$), we now assume that the dual dictionary is feasible ($$z_{\mathcal{N}}^{{\ast}}\geq 0$$) and perform the analogous steps:

Step 1. Check for Optimality. If $$x_{\mathcal{B}}^{{\ast}}\geq 0$$, stop. The current solution is optimal. Note that for the dual simplex method, dual feasibility and complementarity are maintained from the beginning, and the algorithm terminates once a primal feasible solution is discovered.

Step 2. Select Entering Variable. Pick an index $$i \in \mathcal{B}$$ for which x i  ∗  < 0. Variable z i is the entering variable.

Step 3. Compute Dual Step Direction $$\Delta z_{\mathcal{N}}$$. From the dual dictionary, we see that

Step 4. Compute Dual Step Length. We wish to pick the largest s ≥ 0 for which every component of $$z_{\mathcal{N}}$$ remains nonnegative. As in the primal simplex method, this computation involves computing the maximum of some ratios:

If s is not positive, then stop here—the dual is unbounded (implying, of course, that the primal is infeasible).

Step 5. Select Leaving Variable. The leaving variable is chosen as any variable z j , $$j \in \mathcal{N}$$, for which the maximum in the calculation of s is obtained.

Step 6. Compute Primal Step Direction $$\Delta x_{\mathcal{B}}$$. To see how $$x_{\mathcal{B}}^{{\ast}}$$ changes in the dual dictionary, it is convenient to look at the primal dictionary. Since in that dictionary x j is the entering variable, we see that

Step 7. Compute Primal Step Length. Since we know that x i is the leaving variable in the primal dictionary, we see immediately that the step length for the primal variables is

Step 8. Update Current Primal and Dual Solutions. We now have everything we need to update the data in the dictionary:

and

Step 9. Update Basis. Finally, we update the basis:

To further emphasize the similarities between the primal and the dual simplex methods, Figure 6.1 shows the two algorithms side by side.

FIGURE 6.1.

The primal and the dual simplex methods.

5 Two-Phase Methods

Let us summarize the algorithm obtained by applying the dual simplex method as a Phase I procedure followed by the primal simplex method as a Phase II. Initially, we set

Then from (6.1) we see that $$A = \left [\begin{array}{*{10}c} N &B \end{array} \right ]$$, where

and from (6.2) we have

Substituting these expressions into the definitions of $$x_{\mathcal{B}}^{{\ast}}$$, $$z_{\mathcal{N}}^{{\ast}}$$, and ζ  ∗ , we find that

Hence, the initial dictionary reads:

If b has all nonnegative components and $$c_{\mathcal{N}}$$ has all nonpositive components, then this dictionary is optimal—the problem was trivial. Suppose, however, that one of these two vectors (but not both) has components of the wrong sign. For example, suppose that b is okay (all nonnegative components) but $$c_{\mathcal{N}}$$ has some positive components. Then this dictionary is primal feasible, and we can start immediately with the primal simplex method. On the other hand, suppose that $$c_{\mathcal{N}}$$ has all nonpositive components but b has some negative ones. Then the starting dictionary is dual feasible, and we can commence immediately with the dual simplex algorithm.

The last, and most common, case is where both b and $$c_{\mathcal{N}}$$ have components of the wrong sign. In this case, we must employ a two-phase procedure. There are two choices. We could temporarily replace $$c_{\mathcal{N}}$$ with another vector that is nonpositive. Then the modified problem is dual feasible, and so we can apply the dual simplex method to find an optimal solution of this modified problem. After that, the original objective function could be reinstated. With the original objective function, the optimal solution from Phase I is most likely not optimal, but it is feasible, and therefore the primal simplex method can be used to find the optimal solution to the original problem.

The other choice would be to modify b instead of $$c_{\mathcal{N}}$$, thereby obtaining a primal feasible solution to a modified problem. Then we would use the primal simplex method on the modified problem to obtain its optimal solution, which will then be dual feasible for the original problem, and so the dual simplex method can be used to finish the problem.

6 Negative Transpose Property

In our discussion of duality in Chapter 5, we emphasized the symmetry between the primal problem and its dual. This symmetry can be easily summarized by saying that the dual of a standard-form linear programming problem is the negative transpose of the primal problem. Now, in this chapter, the symmetry appears to have been lost. For example, the basis matrix is an m ×m matrix. Why m ×m and not n ×n? It seems strange. In fact, if we had started with the dual problem, added slack variables to it, and introduced a basis matrix on that side it would be an n ×n matrix. How are these two basis matrices related? It turns out that they are not themselves related in any simple way, but the important matrix B  − 1 N is still the negative transpose of the analogous dual construct. The purpose of this section is to make this connection clear.

Consider a standard-form linear programming problem

and its dual

Let w be a vector containing the slack variables for the primal problem, let z be a slack vector for the dual problem, and write both problems in equality form:

and

Introducing three new notations,

the primal problem can be rewritten succinctly as follows:

Similarly, using “hats” for new notations on the dual side,

the dual problem can be rewritten in this way:

Note that the matrix $$\bar{A} = [\begin{array}{cc} A & I \end{array} ]$$ is an m ×(n + m) matrix. The first n columns of it are the initial nonbasic variables and the last m columns are the initial basic columns. After doing some simplex pivots, the basic and nonbasic columns get jumbled up but we can still write the equality

with the understanding that the equality only holds after rearranging the columns appropriately.

On the dual side, the matrix $$\hat{A} = [\begin{array}{cc} - I & {A}^{T} \end{array} ]$$ is an n ×(n + m) matrix. The first n columns of it are the initial basic variables (for the dual problem) and the last m columns are the initial nonbasic columns. If the same set of pivots that were applied to the primal problem are also applied to the dual, then the columns get rearranged in exactly the same way as they did for the primal and we can write

again with the proviso that the columns of one matrix must be rearranged in a specific manner to bring it into exact equality with the other matrix.

Now, the primal dictionary involves the matrix $$\bar{{B}}^{-1}\bar{N}$$ whereas the dual dictionary involves the matrix $$\hat{{B}}^{-1}\hat{N}$$. It probably doesn’t seem at all obvious that these two matrices are negative transposes of each other. To see that it is so, consider what happens when we multiply $$\bar{A}$$ by $$\hat{{A}}^{T}$$ in both the permuted notation and the unpermuted notation:

and

These two expressions obviously must agree so we see that

Putting the two terms on the opposite sides of the equality sign and multiplying on the right by the inverse of $$\hat{{B}}^{T}$$ and on the left by the inverse of $$\bar{B}$$, we get that

which is the property we wished to establish.

Exercises

6.1

Consider the following linear programming problem:

Suppose that, in solving this problem, you have arrived at the following dictionary:

(a)

Which variables are basic? Which are nonbasic?

(b)

Write down the vector, $$x_{\mathcal{B}}^{{\ast}}$$, of current primal basic solution values.

(c)

Write down the vector, $$z_{\mathcal{N}}^{{\ast}}$$, of current dual nonbasic solution values.

(d)

Write down B  − 1 N.

(e)

Is the primal solution associated with this dictionary feasible?

(f)

Is it optimal?

(g)

Is it degenerate?

6.2

Consider the following linear programming problem:

Consider the situation in which x ₃ and x ₅ are basic and all other variables are nonbasic. Write down:

(a)

B,

(b)

N,

(c)

b,

(d)

$$c_{\mathcal{B}}$$,

(e)

$$c_{\mathcal{N}}$$,

(f)

B  − 1 N,

(g)

$$x_{\mathcal{B}}^{{\ast}} = {B}^{-1}b$$,

(h)

$${\zeta }^{{\ast}} = c_{\mathcal{B}}^{T}{B}^{-1}b$$,

(i)

$$z_{\mathcal{N}}^{{\ast}} = {({B}^{-1}N)}^{T}c_{\mathcal{B}}- c_{\mathcal{N}}$$,

(j)

The dictionary corresponding to this basis.

6.3

Solve the problem in Exercise 2.1 using the matrix form of the primal simplex method.

6.4

Solve the problem in Exercise 2.4 using the matrix form of the dual simplex method.

6.5

Solve the problem in Exercise 2.3 using the two-phase approach in matrix form.

6.6

Find the dual of the following linear program:

6.7

Let A be a given m ×n matrix, c a given n-vector, and b a given m-vector. Consider the following max-min problem:

By noting that the inner optimization can be carried out explicitly, show that this problem can be reduced to a linear programming problem. Write it explicitly.

What linear programming problem do you get if the min and max are interchanged?

Notes

In this chapter, we have accomplished two tasks: (1) we have expressed the simplex method in matrix notation, and (2) we have reduced the information we carry from iteration to iteration to simply the list of basic variables together with current values of the primal basic variables and the dual nonbasic variables. In particular, it is not necessary to calculate explicitly all the entries of the matrix B  − 1 N.

What’s in a name? There are times when one thing has two names. So far in this book, we have discussed essentially only one algorithm: the simplex method (assuming, of course, that specific pivot rules have been settled on). But this one algorithm is sometimes referred to as the simplex method and at other times it is referred to as the revised simplex method.  The distinction being made with this new name has nothing to do with the algorithm. Rather it refers to the specifics of an implementation. Indeed, an implementation of the simplex method that avoids explicit calculation of the matrix B  − 1 N is referred to as an implementation of the revised simplex method. We shall see in Chapter 8 why it is beneficial to avoid computing B  − 1 N.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_7

7. Sensitivity and Parametric Analyses

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In this chapter, we consider two related subjects. The first, called sensitivity analysis (or postoptimality analysis) addresses the following question: having found an optimal solution to a given linear programming problem, how much can we change the data and have the current partition into basic and nonbasic variables remain optimal? The second subject addresses situations in which one wishes to solve not just one linear program, but a whole family of problems parametrized by a single real variable.

We shall study parametric analysis in a very specific context in which we wish to find the optimal solution to a given linear programming problem by starting from a problem whose solution is trivially known and then deforming this problem back to the original problem, maintaining as we go optimality of the current solution. The result of this deformation approach to solving a linear programming problem is a new variant of the simplex method, which is called the parametric self-dual simplex method. We will see in later chapters that this variant of the simplex method resembles, in certain respects, the interior-point methods that we shall study.

1 Sensitivity Analysis

One often needs to solve not just one linear programming problem but several closely related problems. There are many reasons that this need might arise. For example, the data that define the problem may have been rather uncertain and one may wish to consider various possible data scenarios. Or perhaps the data are known accurately but change from day to day, and the problem must be resolved for each new day. Whatever the reason, this situation is quite common. So one is led to ask whether it is possible to exploit the knowledge of a previously obtained optimal solution to obtain more quickly the optimal solution to the problem at hand. Of course, the answer is often yes, and this is the subject of this section.

We shall treat a number of possible situations. All of them assume that a problem has been solved to optimality. This means that we have at our disposal the final, optimal dictionary:

Suppose we wish to change the objective coefficients from c to, say, $$\tilde{c}$$. It is natural to ask how the dictionary at hand could be adjusted to become a valid dictionary for the new problem. That is, we want to maintain the current classification of the variables into basic and nonbasic variables and simply adjust ζ  ∗ , $$z_{\mathcal{N}}^{{\ast}}$$, and $$x_{\mathcal{B}}^{{\ast}}$$ appropriately. Recall from (6.​7) to (6.​9) that

Hence, the change from c to $$\tilde{c}$$ requires us to recompute $$z_{\mathcal{N}}^{{\ast}}$$ and ζ  ∗ , but $$x_{\mathcal{B}}^{{\ast}}$$ remains unchanged. Therefore, after recomputing $$z_{\mathcal{N}}^{{\ast}}$$ and ζ  ∗ , the new dictionary is still primal feasible, and so there is no need for a Phase I procedure: we can jump straight into the primal simplex method, and if $$\tilde{c}$$ is not too different from c, we can expect to get to the new optimal solution in a relatively small number of steps.

Now suppose that instead of changing c, we wish to change only the right-hand side b. In this case, we see that we need to recompute $$x_{\mathcal{B}}^{{\ast}}$$ and ζ  ∗ , but $$z_{\mathcal{N}}^{{\ast}}$$ remains unchanged. Hence, the new dictionary will be dual feasible, and so we can apply the dual simplex method to arrive at the new optimal solution fairly directly.

Therefore, changing just the objective function or just the right-hand side results in a new dictionary having nice feasibility properties. What if we need/want to change some (or all) entries in both the objective function and the right-hand side and maybe even the constraint matrix too? In this case, everything changes: ζ  ∗ , $$z_{\mathcal{N}}^{{\ast}}$$, $$x_{\mathcal{B}}^{{\ast}}$$. Even the entries in B and N change. Nonetheless, as long as the new basis matrix B is nonsingular, we can make a new dictionary that preserves the old classification into basic and nonbasic variables. The new dictionary will most likely be neither primal feasible nor dual feasible, but if the changes in the data are fairly small in magnitude, one would still expect that this starting dictionary will get us to an optimal solution in fewer iterations than simply starting from scratch. While there is no guarantee that any of these so-called warm-starts will end up in fewer iterations to optimality, extensive empirical evidence indicates that this procedure often makes a substantial improvement: sometimes the warm-started problems solve in as little as 1% of the time it takes to solve the original problem.

1.1 Ranging

Often one does not wish to solve a modification of the original problem, but instead just wants to ask a hypothetical question:

If I were to change the objective function by increasing or decreasing one of the objective coefficients a small amount, how much could I increase/decrease it without changing the optimality of my current basis?

To study this question, let us suppose that c gets changed to c + tΔc, where t is a real number and Δc is a given vector (which is often all zeros except for a one in a single entry, but we don’t need to restrict the discussion to this case). It is easy to see that $$z_{\mathcal{N}}^{{\ast}}$$ gets incremented by

where

(7.1)

Hence, the current basis will remain dual feasible as long as

(7.2)

We’ve manipulated this type of inequality many times before, and so it should be clear that, for t > 0, this inequality will remain valid as long as

Similar manipulations show that, for t < 0, the lower bound is

Combining these two inequalities, we see that t must lie in the interval

Let us illustrate these calculations with an example. Consider the following linear programming problem:

The optimal dictionary for this problem is given by

The optimal basis is $$\mathcal{B} =\{ 3,\;1,\;5\}$$. Suppose we want to know how much the coefficient of 5 on x ₁ in the objective function can change without altering the optimality of this basis. From the statement of the problem, we see that

Since we are interested in changes in the first coefficient, we put

We partition c according to the final (optimal) basis. Hence, we have

Next, we need to compute $$\Delta z_{\mathcal{N}}$$ using (7.1). We could compute B  − 1 N from scratch, but it is easier to extract it from the constraint coefficients in the final dictionary. Indeed,

Hence, from (7.1) we see that

Now, (7.2) gives the condition on t. Writing it out componentwise, we get

These three inequalities, which must all hold, can be summarized by saying that

Hence, in terms of the coefficient on x ₁, we finally see that it can range from 4. 5 to 6.

Now suppose we change b to b + tΔb and ask how much t can change before the current basis becomes nonoptimal. In this case, $$z_{\mathcal{N}}^{{\ast}}$$ does not change, but $$x_{\mathcal{B}}^{{\ast}}$$ gets incremented by $$t\Delta x_{\mathcal{B}}$$, where

Hence, the current basis will remain optimal as long as t lies in the interval

2 Parametric Analysis and the Homotopy Method

In this section, we illustrate the notion of parametric analysis by applying a technique called the homotopy method to get a new algorithm for solving linear programming problems. The homotopy method is a general technique in which one creates a continuous deformation that changes a given difficult problem into a related but trivially solved problem and then attempts to work backwards from the trivial problem to the difficult problem by solving (hopefully without too much effort) all the problems in between. Of course, there is a continuum of problems between the hard one and the trivial one, and so we shouldn’t expect that this technique will be effective in every situation; but for linear programming and for many other problem domains, it turns out to yield efficient algorithms.

We start with an example. Suppose we wish to solve the following linear programming problem:

The starting dictionary is

This dictionary is neither primal nor dual feasible. Let’s perturb it by adding a positive real number μ to each right-hand side and subtracting it from each objective function coefficient. We now arrive at a family of dictionaries, parametrized by μ:

(7.3)

Clearly, for μ sufficiently large, specifically μ ≥ 3, this dictionary is both primal and dual feasible. Hence, the associated solution $$x = \left [0,0,-1+\mu,-2+\mu,1+\mu \right ]$$ is optimal. Starting with μ large, we reduce it as much as we can while keeping dictionary (7.3) optimal. This dictionary will become nonoptimal as soon as μ < 3, since the associated dual variable $$y_{2}^{{\ast}} = -3+\mu$$ will become negative. In other words, the coefficient of x ₂, which is 3 − μ, will become positive. This change of sign on the coefficient of x ₂ suggests that we make a primal pivot in which x ₂ enters the basis. The usual ratio test (using the specific value of μ = 3) indicates that x ₃ must be the leaving variable. Making the pivot, we get

This dictionary is optimal as long as

These inequalities reduce to

So now we can reduce μ from its current value of 3 down to 4 ∕ 3. If we reduce it below 4 ∕ 3, the primal feasibility inequality, $$-4 + 3\mu \geq 0$$, becomes violated. This violation suggests that we perform a dual pivot with x ₄ serving as the leaving variable. The usual (dual) ratio test (with $$\mu = 4/3$$) then tells us that x ₁ must be the entering variable. Doing the pivot, we get

Now the conditions for optimality are

which reduce to

For the next iteration, we reduce μ to 1 ∕ 2 and see that the inequality that becomes binding is the dual feasibility inequality

Hence, we do a primal pivot with x ₄ entering the basis. The leaving variable is x ₅, and the new dictionary is

For this dictionary, the range of optimality is given by

which reduces to

This range covers μ = 0, and so now we can set μ to 0 and get an optimal dictionary for our original problem:

The algorithm we have just illustrated is called the parametric self-dual simplex method.¹ We shall often refer to it more simply as the self-dual simplex method. It has some attractive features. First, in contrast to the methods presented earlier, this algorithm does not require a separate Phase I procedure. It starts with any problem, be it primal infeasible, dual infeasible, or both, and it systematically performs pivots (whether primal or dual) until it finds an optimal solution.

A second feature is that a trivial modification of the algorithm can avoid entirely ever encountering a degenerate dictionary. Indeed, suppose that, instead of adding/subtracting μ from each of the right-hand sides and objective coefficients, we add/subtract a positive constant times μ. Suppose further that the positive constant is different in each addition/subtraction. In fact, suppose that they are chosen independently from, say, a uniform distribution on $$[1/2,3/2]$$. Then with probability one, the algorithm will produce no primal degenerate or dual degenerate dictionary in any iteration. In Chapter 3, we discussed perturbing the right-hand side of a linear programming problem to avoid degeneracy in the primal simplex method, but back then the perturbation changed the problem. The present perturbation does not in any way affect the problem that is solved.

With the above randomization trick to resolve the degeneracy issue, the analysis of the convergence of the algorithm is straightforward. Indeed, let us consider a problem that is feasible and bounded (the questions regarding feasibility and boundedness are addressed in Exercise 7.10). For each nondegenerate pivot, the next value of μ will be strictly less than the current value. Since each of these μ values is determined by a partition of the variables into basics and nonbasics and there are only a finite number of such partitions, it follows that the method must reach a partition with a negative μ value in a finite number of steps.

3 The Parametric Self-Dual Simplex Method

In the previous section, we illustrated on an example a new algorithm for solving linear programming problems, called the parametric self-dual simplex method. In this section, we shall lay out the algorithm in matrix notation.

Our starting point is an initial dictionary as written in (6.​10) and transcribed here for convenience:

where

Generally speaking, we don’t expect this dictionary to be either primal or dual feasible. So we perturb it by adding essentially arbitrary perturbations $$\bar{x}_{\mathcal{B}}$$ and $$\bar{z}_{\mathcal{N}}$$ to $$x_{\mathcal{B}}^{{\ast}}$$ and $$z_{\mathcal{N}}^{{\ast}}$$, respectively:

We assume that the perturbations are all strictly positive,

so that by taking μ sufficiently large the perturbed dictionary will be optimal. (Actually, to guarantee optimality for large μ, we only need to perturb those primal and dual variables that are negative in the initial dictionary.)

The parametric self-dual simplex method generates a sequence of dictionaries having the same form as the initial one—except, of course, the basis ℬ will change, and hence all the data vectors ($$z_{\mathcal{N}}^{{\ast}}$$, $$\bar{z}_{\mathcal{N}}$$, $$x_{\mathcal{B}}^{{\ast}}$$, and $$\bar{x}_{\mathcal{B}}$$) will change too. Additionally, the current value of the objective function ζ  ∗  will, with the exception of the first dictionary, depend on μ.

One step of the self-dual simplex method can be described as follows. First, we compute the smallest value of μ for which the current dictionary is optimal. Letting μ  ∗  denote this value, we see that

There is either a $$j \in \mathcal{N}$$ for which $$z_{j}^{{\ast}} {+\mu }^{{\ast}}\bar{z}_{j} = 0$$ or an $$i \in \mathcal{B}$$ for which $$x_{i}^{{\ast}} {+\mu }^{{\ast}}\bar{x}_{i} = 0$$ (if there are multiple choices, an arbitrary selection is made). If the blocking constraint corresponds to a nonbasic index $$j \in \mathcal{N}$$, then we do one step of the primal simplex method. If, on the other hand, it corresponds to a basic index $$i \in \mathcal{B}$$, then we do one step of the dual simplex method.

Suppose, for definiteness, that the blocking constraint corresponds to an index $$j \in \mathcal{N}$$. Then, to do a primal pivot, we declare x j to be the entering variable, and we compute the step direction for the primal basic variables as the jth column of the dictionary. That is,

Using this step direction, we find an index $$i \in \mathcal{B}$$ that achieves the maximal value of $$\Delta x_{i}/(x_{i}^{{\ast}} {+\mu }^{{\ast}}\bar{x}_{i})$$. Variable x i is the leaving variable. After figuring out the leaving variable, the step direction vector for the dual nonbasic variables is just the negative of the ith row of the dictionary

After computing the primal and dual step directions, it is easy to see that the step length adjustments are given by

And from these, it is easy to write down the new solution vectors:

Finally, the basis is updated by adding the entering variable and removing the leaving variable

The algorithm is summarized in Figure 7.1.

FIGURE 7.1.

The parametric self-dual simplex method.

Exercises

In solving the following problems, the advanced pivot tool can be used to check your arithmetic: http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​advanced.​html

7.1

The final dictionary for

is

(the last three variables are the slack variables).

(a)

What will be an optimal solution to the problem if the objective function is changed to

(b)

What will be an optimal solution to the problem if the objective function is changed to

(c)

What will be an optimal solution to the problem if the second constraint’s right-hand side is changed to 26?

7.2

For each of the objective coefficients in the problem in Exercise 7.1, find the range of values for which the final dictionary will remain optimal.

7.3

Consider the following dictionary which arises in solving a problem using the self-dual simplex method:

(a)

For which values of μ is the current dictionary optimal?

(b)

For the next pivot in the self-dual simplex method, identify the entering and the leaving variable.

7.4

Solve the linear program given in Exercise 2.3 using the self-dual simplex method. Hint: It is easier to use dictionary notation than matrix notation.

7.5

Solve the linear program given in Exercise 2.4 using the self-dual simplex method. Hint: It is easier to use dictionary notation than matrix notation.

7.6

Solve the linear program given in Exercise 2.6 using the self-dual simplex method. Hint: It is easier to use dictionary notation than matrix notation.

7.7

Using today’s date (MMYY) for the seed value, solve ten problems using the self-dual simplex method:

http://​www.​princeton.​edu/​~rvdb/​JAVA/​pivot/​pd1phase.​html

7.8

Use the self-dual simplex method to solve the following problem:

7.9

Let P μ denote the perturbed primal problem (with perturbation μ). Show that if P μ is infeasible, then P μ′ is infeasible for every μ′ ≤ μ. State and prove an analogous result for the perturbed dual problem.

7.10

Using the notation of Figure 7.1 state precise conditions for detecting infeasibility and/or unboundedness in the self-dual simplex method.

7.11

Consider the following one parameter family of linear programming problems (parametrized by μ):

Starting from μ = ∞, use the parametric simplex method to decrease μ as far as possible. Don’t stop at μ = 0. If you cannot get to $$\mu = -\infty $$, explain why. Hint: the pivots are straight forward and, after the first couple, a clear pattern should emerge which will make the subsequent pivots easy. Clearly indicate the range of μ values for which each dictionary is optimal.

Notes

Parametric analysis has its roots in Gass and Saaty (1955). G.B. Dantzig’s classic book (Dantzig 1963) describes the self-dual simplex method under the name of the self-dual parametric simplex method. It is a special case of “Lemke’s algorithm” for the linear complementarity problem (Lemke 1965) (see Exercise 18.​7). Smale (1983) and Borgwardt (1982) were first to realize that the parametric self-dual simplex method is amenable to probabilistic analysis. For a more recent discussion of homotopy methods and the parametric self-dual simplex method, see Nazareth (1986, 1987).

Bibliography

Borgwardt, K. -H. (1982). The average number of pivot steps required by the simplex-method is polynomial. Zeitschrift für Operations Research, 26, 157–177.

Dantzig, G. (1963). Linear programming and extensions. Princeton: Princeton University Press.Crossref

Gass, S., and Saaty, T. (1955). The computational algorithm for the parametric objective function. Naval Research Logistics Quarterly, 2, 39–45.

Lemke, C. (1965). Bimatrix equilibrium points and mathematical programming. Management Science, 11, 681–689.Crossref

Nazareth, J. (1986). Homotopy techniques in linear programming. Algorithmica, 1, 529–535.Crossref

Nazareth, J. (1987). Computer solutions of linear programs. Oxford: Oxford University Press.

Smale, S. (1983). On the average number of steps of the simplex method of linear programming. Mathematical Programming, 27, 241–262.Crossref

Footnotes

1

In the first edition, this method was called the primal–dual simplex method.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_8

8. Implementation Issues

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In the previous chapter, we rewrote the simplex method using matrix notation. This is the first step toward our aim of describing the simplex method as one would implement it as a computer program. In this chapter, we shall continue in this direction by addressing some important implementation issues.

The most time-consuming steps in the simplex method are the computations

and the difficulty in these steps arises from the B  − 1. Of course, we don’t ever actually compute the inverse of the basis matrix. Instead, we calculate, say, $$\Delta x_{\mathcal{B}}$$ by solving the following system of equations:

(8.1)

where

is the column of N associated with nonbasic variable x j .

Similarly, the calculation of $$\Delta z_{\mathcal{N}}$$ is also broken into two steps:

(8.2)

Here, the first step is the solution of a large system of equations, this time involving B T instead of B, and the second step is the comparatively trivial task of multiplying a vector on the left by the matrix − N T .

Solving the systems of equations (8.1) and (8.2) is where most of the complexity of a simplex iteration lies. We discuss solving such systems in the first two sections. In the second section, we look at the effect of sparsity on these systems. The next few sections explain how to reuse and/or update the computations of one iteration in subsequent iterations. In the final sections, we address a few other issues that affect the efficiency of an implementation.

1 Solving Systems of Equations: LU-Factorization

In this section, we discuss solving systems of equations of the form

where B is an invertible m ×m matrix and b is an arbitrary m-vector. (Analysis of the transpose B T x = b is left to Exercise 8.4.) Our first thought is to use Gaussian elimination. This idea is correct, but to explain how Gaussian elimination is actually implemented, we need to take a fresh look at how it works. To explain, let us consider an example:

(Note that, to emphasize the importance of sparsity, zero entries are simply left blank.) In Gaussian elimination, one begins by subtracting appropriate multiples of the first row from each subsequent row to get zeros in the first column below the diagonal. For our specific example, we subtract 3 ∕ 2 times the first row from the second row and we subtract $$-1/2$$ times the first row from the third row. The result is

Shortly, we will want to remember the values of the nonzero elements in the first column. Therefore, let us agree to do the row operations that are required to eliminate nonzeros, but when we write down the result of the elimination, we will leave the nonzeros there. With this convention, the result of the elimination of the first column can be written as

Note that we have drawn a line to separate the eliminated top/left parts of the matrix from the uneliminated lower-right part.

Next, we eliminate the nonzeros below the second diagonal (there’s only one) by subtracting an appropriate multiple of the second row from each subsequent row. Again, we write the answer without zeroing out the eliminated elements:

After eliminating the third column, we get

Now, the remaining uneliminated part is already an upper triangular matrix, and hence no more elimination is required.

At this point, you are probably wondering how this strangely produced matrix is related to the original matrix B. The answer is both simple and elegant. First, take the final matrix and split it into three matrices: the matrix consisting of all elements on or below the diagonal, the matrix consisting of just the diagonal elements, and the matrix consisting of all elements on or above the diagonal. It is amazing but true that B is simply the product of the resulting lower triangular matrix times the inverse of the diagonal matrix times the upper triangular matrix:

(If you don’t believe it, multiply them and see.) Normally, the product of the lower triangular matrix and the diagonal matrix is denoted by L,

and the upper triangular matrix is denote by U:

The resulting representation,

is called an LU-factorization of B. Finding an LU-factorization is equivalent to Gaussian elimination in the sense that multiplying B on the left by L  − 1 has the effect of applying row operations to B to put it into upper-triangular form U.

The value of an LU-factorization is that it can be used to solve systems of equations. For example, suppose that we wish to solve equation (8.1), where B is as above and

(8.3)

First, we substitute LU for B so that the system becomes

Now, if we let $$y = U\Delta x_{\mathcal{B}}$$, then we can solve

for y, and once y is known, we can solve

for $$\Delta x_{\mathcal{B}}$$. Because L is lower triangular, solving Ly = b is easy. Indeed, writing the system out,

we notice immediately that y ₁ = 7. Then, given y ₁, it becomes clear from the second equation that $$y_{2} = -2 - (3/2)y_{1} = -25/2$$. Continuing in this way, we find that

The process of successively solving for the elements of the vector y starting with the first and proceeding to the last is called forward substitution.

Of course, solving $$U\Delta x_{\mathcal{B}} = y$$ is easy too, since U is upper triangular. The system to solve is given by

(note that, to keep notations simple, we are assuming that the basic indices are 1 through 5 so that $$\Delta x_{\mathcal{B}} = (\Delta x_{1},\Delta x_{2},\Delta x_{3},\Delta x_{4},\Delta x_{5})$$). This time we start with the last equation and see that $$\Delta x_{5} = -1/2$$. Then the second to last equation tells us that $$\Delta x_{4} = 23/2 + 21(\Delta x_{5}) = 1$$. After working our way to the first equation, we have

This process of working from the last element of $$\Delta x_{\mathcal{B}}$$ back to the first is called backward substitution.

2 Exploiting Sparsity

In the previous section, we took a specific matrix B and constructed an LU factorization of it. However, with that example we were lucky in that every diagonal element was nonzero at the moment it was used to eliminate the nonzeros below it. Had we encountered a zero diagonal element, we would have been forced to rearrange the columns and/or the rows of the matrix to put a nonzero element in this position. For a random matrix (whatever that means), the odds of encountering a zero are nil, but a basis matrix can be expected to have plenty of zeros in it, since, for example, it is likely to contain columns associated with slack variables, which are all zero except for one 1. A matrix that contains zeros is called a sparse matrix.

When a sparse matrix has lots of zeros, two things happen. First, the chances of being required to make row and/or column permutations is high. Second, additional computational efficiency can be obtained by making further row and/or column permutations with the aim of keeping L and/or U as sparse as possible.

The problem of finding the “best” permutation is, in itself, harder than the linear programming problem that we ultimately wish to solve. But there are simple heuristics that help to preserve sparsity in L and U. We shall focus on just one such heuristic, called the minimum-degree ordering heuristic, which is describe as follows:

Before eliminating the nonzeros below a diagonal “pivot” element, scan all uneliminated rows and select the sparsest row, i.e., that row having the fewest nonzeros in its uneliminated part (ties can be broken arbitrarily). Swap this row with the pivot row. Then scan the uneliminated nonzeros in this row and select that one whose column has the fewest nonzeros in its uneliminated part. Swap this column with the pivot column so that this nonzero becomes the pivot element. (Of course, provisions should be made to reject such a pivot element if its value is close to zero.)

As a matter of terminology, the number of nonzeros in the uneliminated part of a row/column is called the degree of the row/column. Hence, the name of the heuristic.

Let’s apply the minimum-degree heuristic to the LU-factorization of the matrix B studied in the previous section. To keep track of the row and column permutations, we will indicate original row indices on the left and original column indices across the top. Hence, we start with:

To begin, row 4 has the fewest nonzeros, and within row 4, the − 1 in column 2 belongs to the column with the fewest nonzeros. Hence, we swap rows 1 and 4 and we swap columns 1 and 2 to rewrite B as

Now, we eliminate the nonzeros under the first diagonal element (and, as before, we leave the eliminated nonzeros as they were). The result is

Before doing the elimination associated with the second diagonal element, we note that row 5 is the row with minimum degree, and within row 5, the element 1 in column 3 has minimum column degree. Hence, we swap rows 2 and 5 and we swap columns 1 and 3 to get

Now we eliminate the nonzeros under the second diagonal element to get

For the third stage of elimination, note that row 3 is a minimum-degree row and that, among the nonzero elements of that row, the − 1 is in a minimum-degree column. Hence, for this stage no permutations are needed. The result of the elimination is

For the next stage of the elimination, both of the remaining two rows have the same degree, and hence we don’t need to swap rows. But we do need to swap columns 5 and 4 to put the − 14 into the diagonal position. The result of the swap is

At this point, we notice that the remaining 2 ×2 uneliminated part of the matrix is already upper triangular (in fact, diagonal), and hence no more elimination is needed.

With the elimination completed, we can extract the matrices L and U in the usual way:

and

(Note that the columns of L and the rows of U do not have any “original” indices associated with them, and so no permutation is indicated across the top of L or down the left side of U.)

This LU-factorization has five off-diagonal nonzeros in L and three off-diagonal nonzeros in U for a total of eight off-diagonal nonzeros. In contrast, the LU factorization from the previous section had a total of 12 off-diagonal nonzeros. Hence, the minimum-degree ordering heuristic paid off for this example by reducing the number of nonzeros by 33%. While such a reduction may not seem like a big deal for small matrices such as our 5 ×5 example, for large matrices the difference can be dramatic.

The fact that we have permuted the rows and columns to get this factorization has only a small impact on how one uses the factorization to solve systems of equations. To illustrate, let us solve the same system that we considered before: $$B\Delta x_{\mathcal{B}} = a_{j}$$, where a j is given by (8.3). The first step in solving this system is to permute the rows of a j so that they agree with the rows of L and then to use forward substitution to solve the system Ly = a j . Writing it out, the system looks like this:

The result of the forward substitution is that

(8.4)

The next step is to solve the system $$U\Delta x_{\mathcal{B}} = y$$. Writing this system out, we get

Using backward substitution, we see that

Finally, we rewrite the solution listing the elements of $$\Delta x_{\mathcal{B}}$$ in their original order:

Of course, the answer obtained here agrees with the one obtained at the end of the previous section.

Even with good fill-in minimizing heuristics such as minimum-degree, the LU-factorization remains a significant computational bottleneck. To see why, consider for the moment dense matrices. If we were to write a subroutine to carry out an LU-factorization, we would find that the main body of the routine would have a big triply nested loop:

for each column index j {

for each remaining row index i {

for each remaining column index k {

update the (i,k) entry in accordance with

the aim to make the (i,j) entry be zero

}

}

}

Since each of these loops involves approximately m steps, the LU-factorization routine requires about m ³ operations and hence is called an order m ³ algorithm. Similar considerations tell us that the forward and backward substitutions are both order m ² algorithms. This means that forward and backward substitution can be done much faster than LU-factorization. Indeed, if m = 5, 000, then factorization takes a couple of 1,000 times longer than a forward or backward substitution. Of course, this argument is for dense matrices. But for sparse matrices a similar, if less dramatic, effect is seen. Typically, for sparse matrices, one expects that factorization will take from 10 to 100 times longer than substitution. Therefore, it is important to perform as few LU-factorizations as possible. This is the subject of the next section.

3 Reusing a Factorization

In the previous two sections, we showed how to use an LU-factorization of B to solve the system of equations

for the primal step direction $$\Delta x_{\mathcal{B}}$$. Since the basis matrix doesn’t change much from one iteration of the simplex method to the next (columns get replaced by new ones one at a time), we ask whether the LU-factorization of B from the current iteration might somehow be used again to solve the systems of equations that arise in the next iteration (or even the next several iterations).

Let B denote the current basis (for which a factorization has already been computed) and let $$\tilde{B}$$ denote the basis of the next iteration. Then $$\tilde{B}$$ is simply B with the column that holds the column vector a i associated with the leaving variable x i replaced by a new column vector a j associated with the entering variable x j . This verbal description can be converted into a formula:

(8.5)

Here, as before, e i denotes the vector that is all zeros except for a one in the position associated with index i—to be definite, let us say that this position is the pth position in the vector. To see why this formula is correct, it is helpful to realize that a column vector, say a, times e i T produces a matrix that is all zero except for the pth column, which contains the column vector a.

Since the basis B is invertible, (8.5) can be rewritten as

Denote the matrix in parentheses by E. Recall that $$a_{j} = Ne_{j}$$, since it is the column vector from A associated with the entering variable x j . Hence,

which is a vector we need to compute in the current iteration anyway. Also,

since a i is the column of B associated with the leaving variable x i . Therefore, we can write E more simply as

Now, if E has a simple inverse, then we can use it together with the LU-factorization of B to provide an efficient means of solving systems of equations involving $$\tilde{B}$$. The following proposition shows that E does indeed have a simple inverse.

Proposition 8.1.

Given two column vectors u and v for which 1 + v T u≠0,

Proof.

The proof is trivial. We simply multiply the matrix by its supposed inverse and check that we get the identity:

where the last equality follows from the observation that the parenthesized expression vanishes. □ 

The identity in Proposition 8.1 may seem mysterious, but in fact it has a simple derivation based on the explicit formula for the sum of a geometric series:

This is an identity for real numbers, but it also holds for matrices:

provided that the absolute value of each of the eigenvalues of X is less than one (we don’t prove this here, since it’s just for motivation). Assuming, for the moment, that the absolute values of the eigenvalues of uv T are less than one (actually, all but one of them are zero), we can expand $${(I + u{v}^{T})}^{-1}$$ in a geometric series, reassociate products, and collapse the resulting geometric series to get

where the last equality follows from the fact that $$1/(1 + {v}^{T}u)$$ is a scalar and therefore can be pulled out of the vector/matrix calculation.

Applying Proposition 8.1 to matrix E, we see that

Now, let’s look at the systems of equations that need to be solved in the next iteration. Using tildes to denote items associated with the next iteration, we see that we need to solve

(actually, we should probably put the tilde on the j instead of the a j and on the i instead of the e i , but doing so seems less aesthetically appealing, even though it’s more correct). Recalling that $$\tilde{B} = BE$$, we see that the first system is equivalent to

which can be solved in two stages:

Of course, the second system (involving E) is trivial, since we have an explicit formula for the inverse of E:

(where, in keeping with our tradition, we have used u i to denote the element of u associated with the basic variable x i —that is, u i is the pth entry of u).

The system involving $$\tilde{{B}}^{T}$$ is handled in the same manner. Indeed, first we rewrite it as

and then observe that it too can be solved in two steps:

This time, the first step is the trivial one¹:

Note that the fraction in the preceding equation is a scalar, and so this final expression for u shows that it is a vector with at most two nonzeros—that is, the result is utterly trivial even if the formula looks a little bit cumbersome.

We end this section by returning briefly to our example. Suppose that $$\tilde{B}$$ is B with column 3 replaced by the vector a j given in (8.3). Suppose that

To solve $$\tilde{B}\Delta \tilde{x}_{\mathcal{B}} =\tilde{ a}_{j}$$, we first solve $$Bu =\tilde{ a}_{j}$$ using our LU-factorization of B. The result of the forward and backward substitutions is

Next, we solve for $$\Delta \tilde{x}_{\mathcal{B}} = {E}^{-1}u$$:

Of course, it is easy to check that we have gotten the correct answer: simply multiply $$\tilde{B}$$ times $$\Delta \tilde{x}_{\mathcal{B}}$$ and check that it equals $$\tilde{a}_{j}$$. It does.

4 Performance Tradeoffs

The idea of writing the next basis as a product of the current basis times an easily invertible matrix can be extended over several iterations. For example, if we look k iterations out, we can write

If we have an LU-factorization of B ₀ and we have saved enough information to reconstruct each E j , then we can use this product to solve systems of equations involving B k .

Note that in order to reconstruct E j , all we need to save is the primal step direction vector $$\Delta x_{\mathcal{B}}^{j}$$ (and an integer telling which column it goes in). In actual implementations, these vectors are stored in lists. For historical reasons, this list is called an eta-file (and the matrices E j are called eta matrices). Given the LU-factorization of B ₀ and the eta-file, it is an easy matter to solve systems of equations involving either B or B T . However, as k gets large, the amount of work required to go through the entire eta-file begins to dominate the amount of work that would be required to simply form a new LU-factorization of the current basis. Hence, the best strategy is to use an eta-file but with periodic refactorization of the basis (and accompanied purging of the eta-file).

The question then becomes: how often should one recompute a factorization of the current basis? To answer this question, suppose that we know that it takes F arithmetic operations to form an LU-factorization (of a typical basis for the problem at hand), S operations to do one forward/backward substitution, and E operations to multiply by the inverse of one eta-matrix. Then the number of operations for the initial iteration of the simplex method is F + 2S (since we need to do an LU-factorization and two forward/backward substitutions—one for the system involving the basis and the other for the system involving its transpose). Then, in the next iteration, we need to do two forward/backward substitutions and two eta-inverse calculations. Each subsequent iteration is the same as the previous, except that there are two extra eta-inverse calculations. Hence, the average number of arithmetic operations per iteration if we refactorize after every K iterations is

Treating K as a real variable for the moment, we can differentiate this expression with respect to K, set the derivative equal to zero, and solve for K to get an estimate for the optimal choice of K:

As should be clear from our earlier discussions, E is of order m and, if the basis matrix is dense, F is of order m ³. Hence, for dense matrices, our estimates would indicate that refactorizations should take place every m iterations or so. However, for sparse matrices, F will be substantially less that m ³—more like a constant times m ²—which would indicate that refactorizations should occur on the order of every $$\sqrt{m}$$ iterations. In practice, one typically allows the value of K to be a user-settable parameter whose default value is set to something like 100.

5 Updating a Factorization

There is an important alternative to the eta-matrix method for reusing an LU-factorization, which we shall describe in this section and the next. As always, it is easiest to work with an example, so let’s continue with the same example we’ve been using throughout this chapter.

Recall that the matrix $$\tilde{B}$$ is simply B with its third column replaced by the vector a j given in (8.3):

(Note that we’ve highlighted the new column by putting a box around it.)

Since $${L}^{-1}B = U$$ and $$\tilde{B}$$ differs from B in only one column, it follows that $${L}^{-1}\tilde{B}$$ coincides with U except for the column that got changed. And, since this column got replaced by a j , it follows that this column of $${L}^{-1}\tilde{B}$$ contains $${L}^{-1}a_{j}$$, which we’ve already computed and found to be given by (8.4). Hence,

(8.6)

As we saw before, the columns of L have no “original” indices to relate back to, and so one can simply take them to be numbered in natural order. The same is then true for the rows of L  − 1 and hence for the rows of $${L}^{-1}\tilde{B}$$. That is why the rows shown above are numbered as they are. We’ve shown these numbers explicitly, since they are about to get permuted.

The boxed column in (8.6) is called a spike, since it has nonzeros below the diagonal. The 4 ×4 submatrix constructed from rows 2 through 5 and columns 3, 1, 5, and 4 is called the bump. To get this matrix back into upper-triangular form, one could do row operations to eliminate the nonzeros in the spike that lie below the diagonal. But such row operations could create fill-in anywhere in the bump. Such fill-in would be more than one would like to encounter. However, consider what happens if the spike column is moved to the rightmost column of the bump, shifting the other columns left one position in the process, and if the top row of the bump (i.e., row 2) is moved to the bottom of the bump, shifting the other bump rows up by one. The result of these permutations is

(For future reference, we’ve boxed the bump.) In general, the effect of this permutation is that the column spike gets replaced by a row spike along the bottom row of the bump:

Now any fill-in produced by row operations is confined to the spike row. In our example, there is only one nonzero in the spike row, and to eliminate it we need to add 2 ∕ 7 times row 4 to it. This row operation can be represented algebraically as multiplication on the left by the matrix

That is,

If we denote the new upper triangular matrix by $$\tilde{U}$$, then, solving for $$\tilde{B}$$, we get the following factorization of $$\tilde{B}$$:

We can use this new factorization to solve systems of equations. For example, to solve

we first solve

(8.7)

for y. Then, given y, we compute

and finally we solve

for $$\Delta \tilde{x}_{\mathcal{B}}$$. It is clear from the following chain of equalities that these three steps compute $$\Delta \tilde{x}_{\mathcal{B}}$$:

For our example, we use forward substitution to solve (8.7) for y. The result is

Next, we apply the row operations (in our case, there is only one) to get z:

Finally, backward substitution using $$\tilde{U}$$ is performed to compute $$\Delta \tilde{x}_{\mathcal{B}}$$:

The result of the backward substitution is

which agrees with the solution we got before using eta-matrices.

6 Shrinking the Bump

There is an important enhancement to the factorization updating technique described in the previous section. After permuting rows and columns converting the spike column into a spike row, we can exploit the fact that the spike row is often very sparse (coming as it does from what was originally the top row of the bump) and do further row and column permutations to reduce the size of the bump. To see what we mean, let’s look at our example. First, we note that the leftmost element of the spike row is zero (and hence that the left column of the bump is a singleton column). Therefore, we can simply declare that this column and the corresponding top row do not belong to the bump. That is, we can immediately reduce the size of the bump by one:

This idea can be extended to any column that has a single nonzero in the bump. For example, column 4 is a singleton column too. The trick now is to move this column to the leftmost column of the bump, pushing the intermediate columns to the right, and to apply the same permutation to the rows. After permuting the rows and columns like this, the bump can be reduced in size again:

Furthermore, this reduction in the bump causes a new singleton column to appear (since a singleton need only be a singleton within the bump), namely, column 3. Hence, we permute once again. This time just the column gets permuted, since the singleton is already in the correct row. The bump gets reduced in size once again, now to a 1 ×1 bump, which is not really a bump at all:

Note that we have restored upper triangularity using only permutations; no row operations were needed. While this doesn’t always happen, it is a common and certainly welcome event.

Our example, being fairly small, doesn’t exhibit all the possible bump-reducing permutations. In addition to looking for singleton columns, one can also look for singleton rows. Each singleton row can be moved to the bottom of the bump. At the same time, the associated column is moved to the right-hand column of the bump. After this permutation, the right-hand column and the bottom row can be removed from the bump.

Before closing this section, we reiterate a few important points. First, as the bump gets smaller, the chances of finding further singletons increases. Also, with the exception of the lower-right diagonal element of the bump, all other diagonal elements are guaranteed to be nonzero, since the matrix U from which $$\tilde{U}$$ is derived has this property. Therefore, most bump reductions apply the same permutation to the rows as to the columns. Finally, we have illustrated how to update the factorization once, but this technique can, of course, be applied over and over. Eventually, however, it becomes more efficient to refactorize the basis from scratch.

7 Partial Pricing

In many real-world problems, the number of constraints m is small compared with the number of variables n. Looking over the steps of the primal simplex method, we see that the only steps involving n-vectors are Step 2, in which we pick a nonbasic variable to be the entering variable,

Step 6, in which we compute the step direction for the dual variables,

and Step 8, in which we update the dual variables,

Scanning all the nonbasic indices in Step 2 requires looking at n candidates. When n is huge, this step is likely to be a bottleneck step for the algorithm. However, there is no requirement that all indices be scanned. We could simply scan from the beginning and stop at the first index j for which $$z_{j}^{{\ast}}$$ is negative (as in Bland’s rule). However, in practice, it is felt that picking an index j corresponding to a very negative $$z_{j}^{{\ast}}$$ produces an algorithm that is likely to reach optimality faster. Therefore, the following scheme, referred to as partial pricing is often employed. Initially, scan only a fraction of the indices (say n ∕ 3), and set aside a handful of good ones (say, the 40 or so having the most negative $$z_{j}^{{\ast}}$$). Then use only these 40 in Steps 2, 6, and 8 for subsequent iterations until less than a certain fraction (say, 1 ∕ 2) of them remain eligible. At this point, use (6.​8) to compute the current values of a new batch of n ∕ 3 nonbasic dual variables, and go back to the beginning of this partial pricing process by setting aside the best 40. In this way, most of the iterations look like they only have 40 nonbasic variables. Only occasionally does the grim reality of the full huge number of nonbasic variables surface.

Looking at the dual simplex method (Figure 6.​1), we see that we aren’t so lucky. In it, vectors of length n arise in the max-ratio test:

Here, the entire collection of nonbasic indices must be checked; otherwise, dual feasibility will be lost and the algorithm will fail. Therefore, in cases where n is huge relative to m and partial pricing is used, it is important not to use the dual simplex method as a Phase I procedure. Instead, one should use the technique of adding artificial variables as we did in Chapter 2 to force an initial feasible solution.

8 Steepest Edge

In Chapter 4, we saw that one of the drawbacks of the largest-coefficient rule is its sensitivity to the scale in which variables are quantified. In this section, we shall discuss a pivot rule that partially remedies this problem. Recall that each step of the simplex method is a step along an edge of the feasible region from one vertex to an adjacent vertex. The largest coefficient rule picks the variable that gives the largest rate of increase of the objective function. However, this rate of increase is measured in the “space of nonbasic variables” (we view the basic variables simply as dependent variables). Also, this space changes from one iteration to the next. Hence, in a certain respect, it would seem wiser to measure the rate of increase in the larger space consisting of all the variables, both basic and nonbasic. When the rate of increase is gauged in this larger space, the pivot rule is called the steepest-edge rule. It is the subject of this section.

Fix a nonbasic index $$j \in \mathcal{N}$$. We wish to consider whether x j should be the entering variable. If it were, the step direction vector would be

This vector points out along the edge corresponding to the pivot that would result by letting x j enter the basis. As we know, the objective function is

The derivative of f(x) in the direction of $$\Delta x$$ is given by

The numerator is easy (and familiar):

The denominator is more troublesome:

To calculate $${B}^{-1}Ne_{j}$$ for every $$j \in \mathcal{N}$$ is exactly the same as computing the matrix B  − 1 N, which (as we’ve discussed before) is time consuming and therefore a computation we wish to avoid. But it turns out that we can compute B  − 1 Ne j for every $$j \in \mathcal{N}$$ once at the start (when B is essentially, if not identically, an identity matrix) and then update the norms of these vectors using a simple formula, which we shall now derive.

Let

Suppose that we know these numbers, we use them to perform one step of the simplex method, and we are now at the beginning of the next iteration. As usual, let us denote quantities in this next iteration by putting tildes on them. For example, $$\tilde{B}$$ denotes the new basis matrix. As we’ve seen before, $$\tilde{B}$$ is related to B by the equation $$\tilde{B} = BE$$, where

Now, let’s compute the new ν values:

(8.8)

Recall from (8.2) that we must compute

in the course of the old iteration. If, in addition, we compute

then, expanding out the product in (8.8) and expressing the resulting terms using v and w, we get the following formula for quickly updating the old ν’s to the new ν’s:

Recent computational studies using this update formula have shown that the steepest-edge rule for choosing the entering variable is competitive against, if not superior to, other pivot rules.

Exercises

8.1

(a)

Without permuting rows or columns, compute the LU-factorization of

(8.9)

(b)

Solve the system $$B\Delta x_{\mathcal{B}} = a_{j}$$ where

(c)

Suppose that $$\tilde{B}$$ is B with its second column replaced by a j . Solve the system $$\tilde{B}\Delta \tilde{x}_{\mathcal{B}} =\tilde{ a}_{j}$$ where

using the eta-matrix method.

(d)

Solve the system $$\tilde{B}\Delta \tilde{x}_{\mathcal{B}} =\tilde{ a}_{j}$$ again, this time using the factorization updating method.

8.2

Use the minimum-degree ordering heuristic to find an LU-factorization of the matrix B given by (8.9).

8.3

A permutation matrix is a matrix of zeros and ones for which each row has one 1 and each column has one 1.

(a)

Let B be an m ×m matrix, and let P be a permutation matrix. Show that PB is a matrix obtained by permuting the rows of B and that BP is a matrix obtained by permuting the columns of B. Are the two permutations the same?

(b)

Show that every permutation of the rows of a matrix B corresponds to multiplying B on the left by a permutation matrix.

(c)

Show that for any permutation matrix P,

8.4

Explain how to use the factorization B = LU to solve

Notes

Techniques for exploiting sparsity in matrix factorization have their roots in the paper by Markowitz (1957). A few standard references on matrix factorization are the books of Duff et al. (1986), Golub and VanLoan (1989), and Gill et al. (1991). The eta-matrix technique given in Section 8.3 for using an old basis to solve systems of equations involving the current basis was first described by Dantzig and Orchard-Hayes (1954). The factorization updating technique described in Section 8.5 is the method given by Forrest and Tomlin (1972). The bump reduction techniques of Section 8.6 were first introduced by Saunders (1973) and Reid (1982). The steepest-edge pivoting rule is due to Goldfarb and Reid (1977). A similar rule, known as Devex, was given by Harris (1973).

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Footnotes

1

Occasionally we use the superscript − T for the transpose of the inverse of a matrix. Hence, $${E}^{-T} = {({E}^{-1})}^{T}$$.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_9

9. Problems in General Form

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

Up until now, we have always considered our problems to be given in standard form. However, for real-world problems it is often convenient to formulate problems in the following form:

(9.1)

Two-sided constraints such as those given here are called constraints with ranges. The vector l is called the vector of lower bounds, and u is the vector of upper bounds. We allow some of the data to take infinite values; that is, for each i = 1, 2, …, m,

and, for each $$j = 1,2,\ldots,n$$,

In this chapter, we shall show how to modify the simplex method to handle problems presented in this form.

1 The Primal Simplex Method

It is easiest to illustrate the ideas with an example:

With this formulation, zero no longer plays the special role it once did. Instead, that role is replaced by the notion of a variable or a constraint being at its upper or lower bound. Therefore, instead of defining slack variables for each constraint, we use w i simply to denote the value of the ith constraint:

The constraints can then be interpreted as upper and lower bounds on these variables. Now when we record our problem in a dictionary, we will have to keep explicit track of the upper and lower bound on the original x j variables and the new w i variables. Also, the value of a nonbasic variable is no longer implicit; it could be at either its upper or its lower bound. Hence, we shall indicate which is the case by putting a box around the relevant bound. Finally, we need to keep track of the values of the basic variables. Hence, we shall write our dictionary as follows:

Since all the w i ’s are between their upper and lower bounds, this dictionary is feasible. But it is not optimal, since x ₁ could be increased from its present value at the lower bound, thereby increasing the objective function’s value. Hence, x ₁ shall be the entering variable for the first iteration. Looking at w ₁, we see that x ₁ can be raised only 1 unit before w ₁ hits its lower bound. Similarly, x ₁ can be raised by 4 ∕ 3 units, at which point w ₂ hits its lower bound. Finally, if x ₁ were raised 2 units, then w ₃ would hit its upper bound. The tightest of these constraints is the one on w ₁, and so w ₁ becomes the leaving variable—which, in the next iteration, will then be at its lower bound. Performing the usual row operations, we get

Note, of course, that the objective function value has increased (from − 6 to − 3). For the second iteration, raising x ₂ from its lower bound will produce an increase in ζ. Hence, x ₂ is the entering variable. Looking at the basic variables (x ₁, w ₂, and w ₃), we see that w ₂ will be the first variable to hit a bound, namely, its lower bound. Hence, w ₂ is the leaving variable, which will become nonbasic at its lower bound:

For the third iteration, w ₁ is the entering variable, and w ₃ is the leaving variable, since it hits its upper bound before any other basic variables hit a bound. The result is

Now for the next iteration, note that the coefficients on both w ₃ and w ₂ are positive. But w ₃ is at its upper bound, and so if it were to change, it would have to decrease. However, this would mean a decrease in the objective function. Hence, only w ₂ can enter the basis, in which case x ₂ is the leaving variable getting set to its upper bound:

For this dictionary, both w ₃ and x ₂ are at their upper bounds and have positive coefficients in the formula for ζ. Hence, neither can be moved off from its bound to increase the objective function. Therefore, the current solution is optimal.

2 The Dual Simplex Method

The problem considered in the previous section had an initial dictionary that was feasible. But as always, we must address the case where the initial dictionary is not feasible. That is, we must define a Phase I algorithm. Following the ideas presented in Chapter 5, we base our Phase I algorithm on a dual simplex method. To this end, we need to introduce the dual of (9.1). So first we rewrite (9.1) as

and adding slack variables, we have

We see immediately from the inequality form of the primal that the dual can be written as

(9.2)

Furthermore, at optimality, the dual variables are complementary to the corresponding primal slack variables:

(9.3)

Note that for each i, if b i  > a i , then at optimality v i and q i must be complementary to each other. Indeed, if both were positive, then they could be reduced by an equal amount without destroying feasibility, and the objective function value would strictly decrease, thereby implying that the supposedly optimal solution is not optimal. Similarly, if for some i, b i  = a i , then it is no longer required that v i and q i be complementary at optimality; but, given an optimal solution for which both v i and q i are positive, we can decrease both these values at the same rate until the smaller of the two reaches zero, all the while preserving feasibility of the solution and not changing the objective function value. Hence, there always exists an optimal solution in which every component of v is complementary to the corresponding component of q. The same argument shows that if there exists an optimal solution, then there exists one in which all the components of h and s are complementary to each other as well.

For a real variable ξ, its positive part ξ  +  is defined as

and its negative part ξ  −  is defined similarly as

Clearly, both ξ  +  and ξ  −  are nonnegative. Furthermore, they are complementary,

and their difference represents ξ:

From the complementarity of the components of v against the components of q, we can think of them as the positive and negative parts of the components of just one vector y. So let us write:

Similarly, let us write

If we impose these complementarity conditions not just at optimality but also from the start, then we can eliminate v, q, s, and h from the dual and write it simply as

(9.4)

where the notation y  +  denotes the componentwise positive part of y, etc. This problem is an example from the class of problems called piecewise linear programs. Usually, piecewise linear programs are solved by converting them into linear programs. Here, however, we wish to go in the other direction. We shall present an algorithm for (9.4) that will serve as an algorithm for (9.2). We will call this algorithm the dual simplex method for problems in general form.

To economize on the presentation, we shall present the dual simplex method in the context of a Phase I algorithm for linear programs in general form. Also, to avoid cumbersome notations, we shall present the algorithm with the following example:

(9.5)

The piecewise linear formulation of the dual is

Note that the objective function has coefficients that are infinite. The correct convention is that infinity times a variable is plus infinity if the variable is positive, zero if the variable is zero, and minus infinity if the variable is negative.

Since the objective function is nonlinear (taking positive and negative parts of variables is certainly a nonlinear operation), we will not be able to do the usual row operations on the objective function. Therefore, in each iteration, we simply study it as is. But as usual, we prefer to think in terms of maximization, and so we record the negative of the objective function:

(9.6)

We can of course perform row operations on the two constraints, so we set up the usual sort of dictionary for them:

(9.7)

For the dual problem, all the action takes place at zero. That is, slopes in the objective function change when a variable goes from negative to positive. Since nonbasic variable are supposed to be set where the action is, we associate a current solution with each dictionary by setting the nonbasic variables to zero. Hence, the solution associated with the initial dictionary is

The fact that z ₁ is negative implies that z ₁  −  is a positive number and hence that the objective function value associated with this solution is minus infinity. Whenever the objective function value is minus infinity, we say that the solution is infeasible. We also refer to the associated dictionary as infeasible. Hence, the initial dictionary given in (9.7) is infeasible.

The dual simplex method must start with a dual feasible solution. But since we intend to use the dual simplex method simply to find a feasible solution for (9.5), we are free to change the objective function in (9.5) any way we please. In particular, we can change it from

to

Making that change to the primal leaves the dual objective function unchanged, but produces a feasible dual dictionary:

(9.8)

For comparison purposes, let us also record the corresponding primal dictionary. It is easy to write down the equations defining the w i ’s, but how do we know whether the x j ’s are supposed to be at their upper or their lower bounds? The answer comes from the requirement that the primal and dual satisfy the complementarity conditions given in (9.3). Indeed, from the dual dictionary we see that z ₁ = 1. Hence, $$z_{1}^{+} = 1$$. But since z ₁  +  is just a surrogate for h ₁, we see that h ₁ is positive and hence that g ₁ must be zero. This means that x ₁ must be at its lower bound. Similarly, for the sake of complementarity, x ₂ must also be at its lower bound. Hence, the primal dictionary is

Note that it is infeasible, since w ₁ is not between its upper and lower bounds.

We are now ready to describe the first iteration of the dual simplex method. To this end, we ask whether we can improve the dual objective function value by moving one of the nonbasic variables (y ₁, y ₂, or y ₃) away from zero. Of course, each of these three variables can be moved either to the positive or the negative side of zero; we must analyze these six cases individually. First of all, note that since z ₁ is positive at the current solution, it follows that $$z_{1}^{+} = z_{1}$$ and $$z_{1}^{-} = 0$$ in a neighborhood of the current solution. A similar statement can be made for z ₂, and so we can rewrite (9.6) locally around the current solution as

Now, as y ₁ is increased from zero, the rate of increase of − ξ is simply the derivative of the right-hand side with respect to y ₁, where we must keep in mind that z ₁ and z ₂ are functions of y ₁ via the dictionary (9.8). Hence, the rate of increase is $$-6 - 2 + 1 = -7$$; i.e., the objective function decreases at a rate of 7 units per unit increase of y ₁. If, on the other hand, y ₂ is decreased from zero into negative territory, then the rate of increase of − ξ is the negative of the derivative of the right-hand side. In this case we get no contribution from y ₁  −  but we do get something from z ₁ and z ₂ for a total of $$2 - 1 = 1$$. Hence, the rate of increase as we move in this direction is one unit increase per unit move. We can analyze changes to y ₂ and y ₃. The entire situation can be summarized as follows:

Of these six cases, the only one that brings about an increase in − ξ is the one in which y ₁ is sent negative. Hence, y ₁ shall be our entering variable, and it will go negative. To find the leaving variable, we must ask: as y ₁ goes negative, which of z ₁ and z ₂ will hit zero first? For the current dictionary, z ₂ gets to zero first and so becomes the leaving variable. Performing the usual row operations, the new dictionary for the dual problem is

Let us have a look at the new primal dictionary. The fact that y ₁ was the entering variable in the dual dictionary implies that w ₁ is the leaving variable in the primal. Furthermore, the fact that y ₁ has gone negative implies that y ₁  −  is now positive, and so complementarity then demands that q ₁ be zero; i.e., w ₁ should go to its lower bound. The fact that z ₂ was the leaving variable in the dual dictionary implies that x ₂ is the entering variable in the primal. Hence, the new primal dictionary is

We are now ready to begin the second iteration. Therefore, we ask which nonbasic variable should be moved away from zero (and in which direction). As before, we first note that z ₁ positive implies that $$z_{1}^{+} = z_{1}$$ and $$z_{1}^{-} = 0$$ and that y ₁ negative implies that $$y_{1}^{+} = 0$$ and $$y_{1}^{-} = -y_{1}$$. Hence, the objective function can be written locally around the current solution as

We now summarize the possibilities in a small table:

Note that all the changes are negative, meaning that there are no possibilities to increase the objective function any further. That is, the current dual solution is optimal. Of course, this also could have been deduced by observing that the primal dictionary is feasible (which is what we are looking for, after all).

Even though this example of the dual simplex method has terminated after only one iteration, it should be clear how to proceed had it not terminated.

Now that we have a feasible solution for the primal, we could solve the problem to optimality by simply reinstating the original objective function and proceeding by applying the primal simplex method in a Phase II procedure to find the optimal solution. Since the primal simplex method has already been discussed, we stop here on this problem.

Exercises

Solve the following linear programming problems:

9.1

9.2

Notes

Dantzig (1955) was the first to consider variants of the simplex method that handle bounds and ranges implicitly.

Bibliography

Dantzig, G. (1955). Upper bounds, secondary constraints, and block triangularity in linear programming. Econometrica, 23, 174–183.Crossref

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_10

10. Convex Analysis

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

This book is mostly about linear programming. However, this subject, important as it is, is just a subset of a larger subject called convex analysis. In this chapter, we shall give a brief introduction to this broader subject. In particular, we shall prove a few of the fundamental results of convex analysis and see that their proofs depend on some of the theory of linear programming that we have already developed.

1 Convex Sets

Given a finite set of points, $$z_{1},z_{2},\ldots,z_{n}$$, in $${\mathbb{R}}^{m}$$, a point z in $${\mathbb{R}}^{m}$$ is called a convex combination of these points if¹

where t j  ≥ 0 for each j and $$\sum _{j=1}^{n}t_{j} = 1$$. It is called a strict convex combination if none of the t j ’s vanish. For n = 2, the set of all convex combinations of two points is simply the line segment connecting them.

A subset S of $${\mathbb{R}}^{m}$$ is called convex if, for every x and y in S, S also contains all points on the line segment connecting x and y. That is, $$tx + (1 - t)y \in S$$, for every 0 < t < 1. See Figure 10.1.

FIGURE 10.1.

The set on the left is convex—for any pair of points in the set, the line segment connecting the two points is also contained in the set. The set on the right is not convex—there exists pairs of points, such as the x and y shown, for which the connecting line segment is not entirely in the set.

Certain elementary properties of convex sets are trivial to prove. For example, the intersection of an arbitrary collection of convex sets is convex. Indeed, let S α , α ∈ I, denote a collection of convex sets indexed by some set I. Then the claim is that $$\cap _{\alpha \in I}S_{\alpha }$$ is convex. To see this, consider an arbitrary pair of points x and y in the intersection. It follows that x and y are in each S α . By the convexity of S α it follows that S α contains the line segment connecting x and y. Since each of these sets contains the line segment, so does their intersection. Hence, the intersection is convex.

Here is another easy one:

Theorem 10.1.

A set C is convex if and only if it contains all convex combinations of points in C.

Proof.

Let C be a convex set. By definition, C contains all convex combinations of pairs of points in C. The first nontrivial step is to show that C contains all convex combinations of triples of points in C. To see this, fix z ₁, z ₂, and z ₃ in C and consider

where t j  ≥ 0 for each j and $$\sum _{j=1}^{3}t_{j} = 1$$. If any of the t j ’s vanish, then z is really just a convex combination of two points and so belongs to C. Hence, suppose that each of the t j ’s is strictly positive. Rewrite z as follows:

Since C contains all convex combinations of pairs of points, it follows that

Now, since z is a convex combination of the two points $$\frac{t_{1}} {t_{1}+t_{2}} z_{1} + \frac{t_{2}} {t_{1}+t_{2}} z_{2}$$ and z ₃, both of which belong to C, it follows that z is in C. It is easy to see (pun intended) that this argument can be extended to an inductive proof that C contains all convex combinations of finite collections of points in C. Indeed, one must simply show that the fact that C contains all convex combinations of n points from C implies that it contains all convex combinations of n + 1 points from C. We leave the details to the reader.

Of course, proving that a set is convex if it contains every convex combination of its points is trivial: simply take convex combinations of pairs to get that it is convex.

For each set S in $${\mathbb{R}}^{m}$$ (not necessarily convex), there exists a smallest convex set, which we shall denote by conv(S), containing S. It is defined, quite simply, as the intersection of all convex sets containing S. From our discussion about intersections, it follows that this set is convex. The set conv(S) is called the convex hull of S. This definition can be thought of as a definition from the “outside,” since it involves forming the intersection of a collection of sets that contain S. Our next theorem gives a characterization of convex sets from the “inside”:

Theorem 10.2.

The convex hull conv (S) of a set S in $${\mathbb{R}}^{m}$$ consists precisely of the set of all convex combinations of finite collections of points from S.

Proof.

Let H denote the set of all convex combinations of finite sets of points in S:

It suffices to show that (1) H contains S, (2) H is convex, and (3) every convex set containing S also contains H.

To see that H contains S, just take n = 1 in the definition of H.

To see that H is convex, fix two points x and y in H and a real number 0 < t < 1. We must show that $$z = tx + (1 - t)y \in H$$. The fact that x ∈ H implies that $$x =\sum _{ j=1}^{r}p_{j}x_{j}$$, for some r ≥ 1, where p j  > 0 for $$j = 1,2,\ldots,r$$, $$\sum _{j=1}^{r}p_{j} = 1$$, and x j  ∈ S for $$j = 1,2,\ldots,r$$. Similarly, the fact that y is in H implies that $$y =\sum _{ j=1}^{s}q_{j}y_{j}$$, for some s ≥ 1, where q j  > 0 for $$j = 1,2,\ldots,s$$, $$\sum _{j=1}^{s}q_{j} = 1$$, and y j  ∈ S for $$j = 1,2,\ldots,s$$. Hence,

Since the coefficients $$(tp_{1},\ldots,tp_{r},(1 - t)q_{1},\ldots,(1 - t)q_{s})$$ are all positive and sum to one, it follows that this last expression for z is a convex combination of r + s points from S. Hence, z is in H. Since x and y were arbitrary points in H and t was an arbitrary real number between zero and one, the fact that z ∈ H implies that H is convex.

It remains simply to show that H is contained in every convex set containing S. Let C be such a set (i.e., convex and containing S). From Theorem 10.1 and the fact that C contains S, it follows that C contains all convex combinations of points in S. Hence, C contains H.

2 Carathéodory’s Theorem

In the previous section, we showed that the convex hull of a set S can be constructed by forming all convex combinations of finite sets of points from S. In 1907, Carathéodory showed that it is not necessary to use all finite sets. Instead, m + 1 points suffice:

THEOREM 10.3.

The convex hull conv (S) of a set S in $${\mathbb{R}}^{m}$$ consists of all convex combinations of m + 1 points from S:

PROOF.

Let H denote the set on the right. From Theorem 10.2, we see that H is contained in conv(S). Therefore, it suffices to show that every point in conv(S) belongs to H. To this end, fix a point z in conv(S). By Theorem 10.2, there exists a collection of, say, n points $$z_{1},z_{2},\ldots,z_{n}$$ in S and associated nonnegative multipliers $$t_{1},t_{2},\ldots,t_{n}$$ summing to one such that

(10.1)

Let A denote the matrix consisting of the points $$z_{1},z_{2},\ldots,z_{n}$$ as the columns of A:

Also, let x  ∗  denote the vector consisting of the multipliers $$t_{1},t_{2},\ldots,t_{n}$$:

Finally, let b = z. Then from (10.1), we see that x  ∗  is feasible for the following linear programming problem:

(10.2)

The fundamental theorem of linear programming (Theorem 3.4) tells us that every feasible linear program has a basic feasible solution. For such a solution, only the basic variables can be nonzero. The number of basic variables in (10.2) coincides with the number of equality constraints; that is, there are at most m + 1 variables that are nonzero. Hence, this basic feasible solution corresponds to a convex combination of just m + 1 of the original n points. (See Exercise 10.5.)

It is easy to see that the number m + 1 is the best possible. For example, the point $$(1/(m + 1),1/(m + 1),\ldots,1/(m + 1))$$ in ℝ m belongs to the convex hull of the m + 1 points $$e_{1},e_{2},\ldots,e_{m},0$$ but is not a convex combination of any subset of them.

3 The Separation Theorem

We shall define a halfspace of $${\mathbb{R}}^{n}$$ to be any set given by a single (nontrivial) linear inequality:

(10.3)

Every halfspace is convex. To see this, suppose that $$x = (x_{1},x_{2},\ldots,x_{n})$$ and $$y = (y_{1},y_{2},\ldots,y_{n})$$ both satisfy the linear inequality in (10.3). Fix t between zero and one. Then both t and 1 − t are nonnegative, and so multiplying by them preserves the direction of inequality. Therefore, multiplying $$\sum _{j}a_{j}x_{j} \leq b$$ by t and $$\sum _{j}a_{j}y_{j} \leq b$$ by 1 − t and then adding, we get

That is, $$tx + (1 - t)y$$ also satisfies the inequality defining the halfspace.

If we allow the vector of coefficients $$(a_{1},a_{2},\ldots,a_{n})$$ in the definition of a halfspace to vanish, then we call the set so defined a generalized halfspace. It is easy to see that every generalized halfspace is simply a halfspace, all of $${\mathbb{R}}^{n}$$, or the empty set. Also, every generalized halfspace is clearly convex.

A polyhedron is defined as the intersection of a finite collection of generalized halfspaces. That is, a polyhedron is any set of the form

Every polyhedron, being the intersection of a collection of convex sets, is convex.

The following theorem is called the Separation Theorem for polyhedra.

THEOREM 10.4.

Let P and $$\tilde{P}$$ be two disjoint nonempty polyhedra in $${\mathbb{R}}^{n}$$ . Then there exist disjoint halfspaces H and $$\tilde{H}$$ such that P ⊂ H and $$\tilde{P} \subset \tilde{ H}$$ .

PROOF.

Suppose that P and $$\tilde{P}$$ are given by the following systems of inequalities:

The disjointness of P and $$\tilde{P}$$ implies that there is no solution to the system

(10.4)

To continue the proof, we need a result known as Farkas’ Lemma, which says that Ax ≤ b has no solutions if and only if there is an m-vector y such that

We shall prove this result in the next section. For now, let us apply Farkas’ Lemma to the situation at hand. Indeed, the fact that there are no solutions to (10.4) implies that there exists a vector, which we shall write in block form as

such that

(10.5)

(10.6)

(10.7)

From the last condition, we see that either b T y < 0 or $$\tilde{{b}}^{T}\tilde{y} < 0$$ (or both). Without loss of generality, we may assume that

Farkas’ Lemma (this time applied in the other direction) together with the nonemptiness of P now implies that

Put

These sets are clearly halfspaces. To finish the proof, we must show that they are disjoint and contain their corresponding polyhedra.

First of all, it follows from (10.7) that H and $$\tilde{H}$$ are disjoint. Indeed, suppose that x ∈ H. Then $${({A}^{T}y)}^{T}x \leq {b}^{T}y < -\tilde{{b}}^{T}\tilde{y}$$, which implies that x is not in $$\tilde{H}$$.

To show that P ⊂ H, fix x in P. Then Ax ≤ b. Since y ≥ 0 (as we know from (10.6)), it follows then that $${y}^{T}Ax \leq {y}^{T}b$$. But this is exactly the condition that says that x belongs to H. Since x was an arbitrary point in P, it follows that P ⊂ H.

Showing that $$\tilde{P}$$ is a subset of $$\tilde{H}$$ is similar. Indeed, suppose that $$x \in \tilde{ P}$$. Then $$\tilde{A}x \leq \tilde{ b}$$. Multiplying on the left by $$-\tilde{{y}}^{T}$$ and noting that $$\tilde{y} \geq 0$$, we see that $$-\tilde{{y}}^{T}\tilde{A}x \geq -\tilde{{y}}^{T}\tilde{b}$$. But from (10.5) we see that $$-\tilde{{y}}^{T}\tilde{A}x = {y}^{T}Ax$$, and so this last inequality is exactly the condition that $$x \in \tilde{ H}$$. Again, the arbitrariness of $$x \in \tilde{ P}$$ implies that $$\tilde{P} \subset \tilde{ H}$$, and the proof is complete.

4 Farkas’ Lemma

The following result, known as Farkas’ Lemma, played a fundamental role in the proof of the separation theorem of the preceding section (Theorem 10.4). In this section, we state it formally as a lemma and give its proof.

Lemma 1.

The system Ax ≤ b has no solutions if and only if there is a y such that

(10.8)

Proof.

Consider the linear program

and its dual

Clearly, the dual is feasible (just take y = 0). So if the primal is feasible, then the dual is bounded. Also, if the primal is infeasible, the dual must be unbounded. That is, the primal is infeasible if and only if the dual is unbounded. To finish the proof, we claim that the dual is unbounded if and only if there exists a solution to (10.8). Indeed, suppose that the dual is unbounded. The dual simplex method is guaranteed to prove that it is unbounded, and it does so as follows. At the last iteration, a step direction $$\Delta y$$ is computed that preserves feasibility, i.e.,

is a descent direction for the objective function, i.e.,

and is a direction for which the step length is unbounded, i.e.,

But these three properties show that $$\Delta y$$ is the solution to (10.8) that we were looking for. Conversely, suppose that there is a solution to (10.8). Call it $$\Delta y$$. It is easy to see that starting from y = 0, this step direction provides an unbounded decrease in the objective function. This completes the proof.

5 Strict Complementarity

In this section, we consider the usual inequality-form linear programming problem, which we write with its slack variables shown explicitly:

(10.9)

As we know, the dual can be written as follows:

(10.10)

In our study of duality theory in Chapter 5, we saw that every pair of optimal solutions to these two problems possesses a property called complementary slackness. If (x  ∗ , w  ∗ ) denotes an optimal solution to the primal and $$({y}^{{\ast}},{z}^{{\ast}})$$ denotes an optimal solution to the dual, then the complementary slackness theorem says that, for each j = 1, 2, …, n, either $$x_{j}^{{\ast}} = 0$$ or z j  ∗  = 0 (or both) and, for each i = 1, 2, …, m, either $$y_{i}^{{\ast}} = 0$$ or w i  ∗  = 0 (or both). In this section, we shall prove that there are optimal pairs of solutions for which the parenthetical “or both” statements don’t happen. That is, there are optimal solutions for which exactly one member of each pair $$(x_{j}^{{\ast}},z_{j}^{{\ast}})$$ vanishes and exactly one member from each pair $$(y_{i}^{{\ast}},w_{i}^{{\ast}})$$ vanishes. In such cases, we say that the optimal solutions are strictly complementary to each other. The strictness of the complementary slackness is often expressed by saying that x  ∗  + z  ∗  > 0 and y  ∗  + w  ∗  > 0.²

As a warm-up, we prove the following theorem.

Theorem 10.5.

If both the primal and the dual have feasible solutions, then there exists a primal feasible solution $$(\bar{x},\bar{w})$$ and a dual feasible solution $$(\bar{y},\bar{z})$$ such that $$\bar{x} +\bar{ z} > 0$$ and $$\bar{y} +\bar{ w} > 0$$ .

Proof.

If there is a feasible primal solution $$\bar{x}$$ for which $$\bar{x}_{j} > 0$$, then it doesn’t matter whether there is a feasible dual solution whose jth slack variable is strictly positive. But what about indices j for which x j  = 0 for every feasible solution? Let j be such an index. Consider the following linear programming problem:

(10.11)

This problem is feasible, since its constraints are the same as for the original primal problem (10.9). Furthermore, it has an optimal solution (the corresponding objective function value is zero). The dual of (10.11) is:

By the strong duality theorem, the dual has an optimal solution, say y′. Letting z′ denote the corresponding slack variable, we have that

Now, let y be any feasible solution to (10.10) and let z be the corresponding slack variable. Then the above properties of y′ and z′ imply that y + y′ is feasible for (10.10) and its slack is $$z + z^{\prime} + e_{j}$$. Clearly, for this dual feasible solution we have that the jth component of its vector of slack variables is at least 1. To summarize, we have shown that, for each j, there exists a primal feasible solution, call it (x (j), w (j)), and a dual feasible solution, call it (y (j), z (j)), such that $$x_{j}^{(j)} + z_{j}^{(j)} > 0$$. In the same way, one can exhibit primal and dual feasible solutions for which each individual dual variable and its corresponding primal slack add to a positive number. To complete the proof, we now form a strict convex combination of these n + m feasible solutions. Since the feasible region for a linear programming problem is convex, these convex combinations preserve primal and dual feasibility. Since the convex combination is strict, it follows that every primal variable and its dual slack add to a strictly positive number as does every dual variable and its primal slack.

A variable x j that must vanish in order for a linear programming problem to be feasible is called a null variable. The previous theorem says that if a variable is null, then its dual slack is not null.

The following theorem is called the Strict Complementary Slackness Theorem

Theorem 10.6.

If a linear programming problem has an optimal solution, then there is an optimal solution (x ∗ ,w ∗ ) and an optimal dual solution (y ∗ ,z ∗ ) such that x ∗ + z ∗ > 0 and y ∗ + w ∗ > 0.

We already know from the complementary slackness theorem (Theorem 5.​1) that x  ∗  and z  ∗  are complementary to each other as are y  ∗  and w  ∗ . This theorem then asserts that the complementary slackness is strict.

Proof.

The proof is much the same as the proof of Theorem 10.5 except this time we look at an index j for which x j vanishes in every optimal solution. We then consider the following problem:

(10.12)

where ζ  ∗  denotes the objective value of the optimal solution to the original problem. In addition to the dual variables y corresponding to the Ax ≤ b constraints, there is one more dual variable, call it t, associated with the constraint c T x ≥ ζ  ∗ . The analysis of problem (10.12) is similar to the analysis given in Theorem 10.5 except that one must now consider two cases: (a) the optimal value of t is strictly positive and (b) the optimal value of t vanishes. The details are left as an exercise (see Exercise 10.6).

Exercises

10.1

Is $${\mathbb{R}}^{n}$$ a polyhedron?

10.2

For each $$b \in {\mathbb{R}}^{m}$$, let ξ  ∗ (b) denote the optimal objective function value for the following linear program:

Suppose that ξ  ∗ (b) < ∞ for all b. Show that the function ξ  ∗ (b) is concave (a function f on $${\mathbb{R}}^{m}$$ is called concave if $$f(tx + (1 - t)y) \geq tf(x) + (1 - t)f(y)$$ for all x and y in $${\mathbb{R}}^{m}$$ and all 0 < t < 1). Hint: Consider the dual problem.

10.3

Describe how one needs to modify the proof of Theorem 10.4 to get a proof of the following result:

Let P and $$\tilde{P}$$ be two disjoint polyhedra in $${\mathbb{R}}^{n}$$. Then there exist disjoint generalized halfspaces H and $$\tilde{H}$$ such that P ⊂ H and $$\tilde{P} \subset \tilde{ H}$$.

10.4

Find a strictly complementary solution to the following linear programming problem and its dual:

10.5

There is a slight oversimplification in the proof of Theorem 10.3. Can you spot it? Can you fix it?

10.6

Complete the proof of Theorem 10.6.

10.7

Interior solutions. Prove the following statement: If a linear programming problem has feasible solutions and the set of feasible solutions is bounded, then there is a strictly positive dual feasible solution: y > 0 and z > 0. Hint. It is easier to prove the equivalent statement: if a linear programming problem has feasible solutions and the dual has null variables, then the set of primal feasible solutions is an unbounded set.

Notes

Carathéodory (1907) proved Theorem 10.3. Farkas (1902) proved Lemma 1. Several similar results were discovered by many others, including Gordan (1873), Stiemke (1915), Ville (1938), and Tucker (1956). The standard reference on convex analysis is Rockafellar (1970).

Bibliography

Carathéodory, C. (1907). Über den variabilitätsbereich der koeffizienten von potenzreihen, die gegebene werte nicht annehmen. Mathematische Annalen, 64, 95–115.Crossref

Farkas, J. (1902). Theorie der einfachen Ungleichungen. Journal für die reine und angewandte Mathematik, 124, 1–27.

Gordan, P. (1873). Über die Auflösung linearer Gleichungen mit reelen coefficienten. Mathematische Annalen, 6, 23–28.Crossref

Rockafellar, R. (1970). Convex analysis. Princeton: Princeton University Press.Crossref

Stiemke, E. (1915). Über positive Lösungen homogener linearer Gleichungen. Mathematische Annalen, 76, 340–342.Crossref

Tucker, A. (1956). Dual systems of homogeneous linear equations. Annals of Mathematics Studies, 38, 3–18.

Ville, J. (1938). Sur la théorie général des jeux ou intervient l’habileté des jouers. In E. Borel (Ed.), Traité du Calcul des probabilités et des ses applications. Paris: Gauthiers-Villars.

Footnotes

1

Until now we’ve used subscripts for the components of a vector. In this chapter, subscripts will be used to list sequences of vectors. Hopefully, this will cause no confusion.

2

Given any vector ξ, we use the notation ξ > 0 to indicate that every component of ξ is strictly positive: ξ j  > 0 for all j.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_11

11. Game Theory

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In this chapter, we shall study if not the most practical then certainly an elegant application of linear programming. The subject is called game theory, and we shall focus on the simplest type of game, called the finite two-person zero-sum game, or just matrix game for short. Our primary goal shall be to prove the famous Minimax Theorem, which was first discovered and proved by John von Neumann in 1928. His original proof of this theorem was rather involved and depended on another beautiful theorem from mathematics, the Brouwer Fixed-Point Theorem. However, it eventually became clear that the solution of matrix games could be found by solving a certain linear programming problem and that the Minimax Theorem is just a fairly straightforward consequence of the Duality Theorem.

1 Matrix Games

A matrix game is a two-person game defined as follows. Each person first selects, independently of the other, an action from a finite set of choices (the two players in general will be confronted with different sets of actions from which to choose). Then both reveal to each other their choice. If we let i denote the first player’s choice and j denote the second player’s choice, then the rules of the game stipulate that the first player will pay the second player a ij dollars. The array of possible payments

is presumed known to both players before the game begins. Of course, if a ij is negative for some pair (i, j), then the payment goes in the reverse direction—from the second player to the first. For obvious reasons, we shall refer to the first player as the row player and the second player as the column player. Since we have assumed that the row player has only a finite number of actions from which to choose, we can enumerate these actions and assume without loss of generality that i is simply an integer selected from 1 to m. Similarly, we can assume that j is simply an index ranging from 1 to n (in its real-world interpretation, row action 3 will generally have nothing to do with column action 3—the number 3 simply indicates that it is the third action in the enumerated list of choices).

Let us look at a specific familiar example. Namely, consider the game every child knows, called Paper–Scissors–Rock. To refresh the memory of older readers, this is a two-person game in which at the count of three each player declares either Paper, Scissors, or Rock. If both players declare the same object, then the round is a draw. But Paper loses to Scissors (since scissors can cut a piece of paper), Scissors loses to Rock (since a rock can dull scissors), and finally Rock loses to Paper (since a piece of paper can cover up a rock—it’s a weak argument but that’s the way the game is defined). Clearly, for this game, if we enumerate the actions of declaring Paper, Scissors, or Rock as 1, 2, 3, respectively, then the payoff matrix is

With this matrix, neither player has an obvious (i.e., deterministic) winning strategy. If the column player were always to declare Paper (hoping that the row player will declare Rock), then the row player could counter by always declaring Scissors and guaranteeing herself a winning of one dollar in every round. In fact, if the column player were to stick to any specific declaration, then the row player would eventually get wise to it and respond appropriately to guarantee that she wins. Of course, the same logic applies to the row player. Hence, neither player should employ the same declaration over and over. Instead, they should randomize their declarations. In fact, due to the symmetry of this particular game, both players should make each of the three possible declarations with equal likelihood.

But what about less trivial games? For example, suppose that the payoffs in the Paper–Scissors–Rock game are altered so that the payoff matrix becomes

This new game still has the property that every deterministic strategy can be foiled by an intelligent opponent. Hence, randomized behavior remains appropriate. But the best probabilities are no longer uniformly 1 ∕ 3. Also, who has the edge in this game? Since the total of the payoffs that go from the row player to the column player is 10 whereas the total of the payoffs that go to the row player is 11, we suspect that the row player might have the edge. But this is just a guess. Is it correct? If it is correct, how much can the row player expect to win on average in each round? If the row player knows this number accurately and the column player does not, then the row player could offer to pay the column player a small fee for playing each round. If the fee is smaller than the expected winnings, then the row player can still be confident that over time she will make a nice profit. The purpose of this chapter is to answer these questions precisely.

Let us return now to the general setup. Consider the row player. By a randomized strategy, we mean that, at each play of the game, it appears (from the column player’s viewpoint) that the row player is making her choices at random according to some fixed probability distribution. Let y i denote the probability that the row player selects action i. The vector y composed of these probabilities is called a stochastic vector. Mathematically, a vector is a stochastic vector if it has nonnegative components that sum up to one:

where e denotes the vector consisting of all ones. Of course, the column player must also adopt a randomized strategy. Let x j denote the probability that the column player selects action j, and let x denote the stochastic vector composed of these probabilities.

The expected payoff to the column player is computed by summing over all possible outcomes the payoff associated with that outcome times the probability of the outcome. The set of possible outcomes is simply the set of pairs (i, j) as i ranges over the row indices ($$1,2,\ldots,m$$) and j ranges over the column indices ($$1,2,\ldots,n$$). For outcome (i, j) the payoff is a ij , and, assuming that the row and column players behave independently, the probability of this outcome is simply $$y_{i}x_{j}$$. Hence, the expected payoff to the column player is

2 Optimal Strategies

Suppose that the column player adopts strategy x (i.e., decides to play in accordance with the stochastic vector x). Then the row player’s best defense is to use the strategy y  ∗  that achieves the following minimum:

(11.1)

From the fundamental theorem of linear programming, we know that this problem has a basic optimal solution. For this problem, the basic solutions are simply y vectors that are zero in every component except for one, which is one. That is, the basic optimal solutions correspond to deterministic strategies. This is fairly obvious if we look again at our example. Suppose that

Then

and so the row player’s best choice is to select either i = 1 (Paper) or i = 3 (Rock) or any combination thereof. That is, an optimal solution is y  ∗  = (1, 0, 0) (it is not unique).

Since for any given x the row player will adopt the strategy that achieves the minimum in (11.1), it follows that the column player should employ a strategy x  ∗  that attains the following maximum:

(11.2)

where the max and the min are over all stochastic vectors (of the appropriate dimension).

The question then becomes: how do we solve (11.2)? It turns out that this problem can be reformulated as a linear programming problem. Indeed, we have already seen that the inner optimization (the minimization) can be taken over just the deterministic strategies:

where we have used e i to denote the vector of all zeros except for a one in position i. Hence, the max-min problem given in (11.2) can be rewritten as

Now, if we introduce a new variable, v, representing a lower bound on the $$e_{i}^{T}Ax$$’s, then we see that the problem can be recast as a linear program:

Switching back to vector notation, the problem can be written as

Finally, writing in block-matrix form, we get

(11.3)

Now let’s turn it around. By symmetry, the row player seeks a strategy y  ∗  that attains optimality in the following min-max problem:

which can be reformulated as the following linear program:

Writing in block-matrix form, we get

(11.4)

3 The Minimax Theorem

Having reduced the computation of the optimal strategies x  ∗  and y  ∗  to the solution of linear programs, it is now a simple matter to show that they are consistent with each other. The next theorem, which establishes this consistency, is called the Minimax Theorem:

Theorem 11.1.

There exist stochastic vectors x ∗ and y ∗ for which

Proof.

The proof follows trivially from the observation that (11.4) is the dual of (11.3). Therefore, $${v}^{{\ast}} = {u}^{{\ast}}$$. Furthermore,

and similarly,

 □ 

The common optimal value $${v}^{{\ast}} = {u}^{{\ast}}$$ of the primal and dual linear programs is called the value of the game. From the Minimax Theorem, we see that, by adopting strategy y  ∗ , the row player assures herself of losing no more than v units per round on the average. Similarly, the column player can assure himself of winning at least v units per round on the average by adopting strategy x  ∗ . A game whose value is zero is therefore a fair game. Games where the roles of the two players are interchangeable are clearly fair. Such games are called symmetric. They are characterized by payoff matrices having the property that $$a_{ij} = -a_{ji}$$ for all i and j (in particular, m must equal n and the diagonal must vanish).

For the Paper–Scissors–Rock game, the linear programming problem that the column player needs to solve is

In nonmatrix notation, it looks like this:

This linear programming problem deviates from our standard inequality form in two respects: (1) it has an equality constraint and (2) it has a free variable. There are several ways in which one can convert this problem into standard form. The most compact conversion is as follows. First, use the equality constraint to solve explicitly for one of the x j ’s, say x ₃:

Then eliminate this variable from the remaining equations to get

The elimination of x ₃ has changed the last constraint from an equality into an inequality.

The next step is to write down a starting dictionary. To do this, we need to introduce slack variables for each of these constraints. It is natural (and desirable) to denote the slack variable for the last constraint by x ₃. In fact, doing this, we get the following starting dictionary:

The variable v is not constrained to be nonnegative. Therefore, there is no reason for it to be nonbasic. Let us do an arbitrary pivot with v as the entering variable and any basic variable as the leaving variable (well, not exactly any—we must make sure that it causes no division by 0, so therefore x ₃ is not a candidate). Picking x ₄ to leave, we get

Since v is free of sign constraints, it will never leave the basis (since a leaving variable is, by definition, a variable that hits its lower bound—v has no such bound). Therefore, we may as well remove it from the dictionary altogether; it can always be computed at the end. Hence, we note that

or better yet that

and the dictionary now becomes

At last, we are in a position to apply the simplex method. Two (tedious) iterations bring us to the optimal dictionary. Since it involves fractions, we multiply each equation by an integer to make each number in the dictionary an integer. Indeed, after multiplying by 102, the optimal dictionary is given by

From this dictionary, it is easy to read off the optimal primal solution:

Also, since x ₄, x ₅, and x ₆ are complementary to y ₁, y ₂, and y ₃ in the dual problem, the optimal dual solution is

Finally, the value of the game is

which indicates that the row player does indeed have an advantage and can expect to make on the average close to 16 cents per round.

4 Poker

Some card games such as poker involve a round of bidding in which the players at times bluff by increasing their bid in an attempt to coerce their opponents into backing down, even though if the challenge is accepted they will surely lose. Similarly, they will sometimes underbid to give their opponents false hope. In this section, we shall study a simplified version of poker (the real game is too hard to analyze) to see if bluffing and underbidding are justified bidding strategies.

Simplified poker involves two players, A and B, and a deck having three cards, 1, 2, and 3. At the beginning of a round, each player “antes up” $1 and is dealt one card from the deck. A bidding session follows in which each player in turn, starting with A, either (a) bets and adds $1 to the “kitty” or (b) passes. Bidding terminates when

a bet is followed by a bet,

a pass is followed by a pass, or

a bet is followed by a pass.

In the first two cases, the winner of the round is decided by comparing cards, and the kitty goes to the player with the higher card. In the third case, bet followed by pass, the player who bet wins the round independently of who had the higher card (in real poker, the player who passes is said to fold).

With these simplified betting rules, there are only five possible betting scenarios:

A passes,

B passes:

$1 to holder of higher card

A passes,

B bets,

A passes:

$1 to B

A passes,

B bets,

A bets:

$2 to holder of higher card

A bets,

B passes:

$1 to A

A bets,

B bets:

$2 to holder of higher card

After being dealt a card, player A will decide to bet along one of three lines:

1.

Pass. If B bets, pass again.

2.

Pass. If B bets, bet.

3.

Bet.

Similarly, after being dealt a card, player B can bet along one of four lines:

1.

Pass no matter what.

2.

If A passes, pass, but if A bets, bet.

3.

If A passes, bet, but if A bets, pass.

4.

Bet no matter what.

To model the situation as a matrix game, we must identify each player’s pure strategies. A pure strategy is a statement of what line of betting a player intends to follow for each possible card that the player is dealt. Hence, the players’ pure strategies can be denoted by triples $$(y_{1},y_{2},y_{3})$$, where y i is the line of betting that the player will use when holding card i. (For player A, the y i ’s can take values 1, 2, and 3, whereas for player B, they can take values 1, 2, 3, and 4.)

Given a pure strategy for both players, one can compute the average payment from, say, A to B. For example, suppose that player A adopts strategy (3, 1, 2) and player B adopts strategy (3, 2, 4). There are six ways in which the cards can be dealt, and we can analyze each of them as follows:

Since each of the six deals are equally likely, the average payment from A to B is $$(2 + 2 + 1 + 1 - 2 - 1)/6 = 0.5$$.

The calculation of the average payment must be carried out for every combination of pairs of strategies. How many are there? Player A has 3 ×3 ×3 = 27 pure strategies and player B has 4 ×4 ×4 = 64 pure strategies. Hence, there are 27 ×64 = 1, 728 pairs. Calculating the average payment for all these pairs is a daunting task. Fortunately, we can reduce the number of pure strategies (and hence the number of pairs) that need to be considered by making a few simple observations.

The first observation is that a player holding a 1 should never answer a bet with a bet, since the player will lose regardless of the answering bet and will lose less by passing. This logic implies that, when holding a 1,

player A should refrain from betting along line 2;

player B should refrain from betting along lines 2 and 4.

More clearly improvable strategies can be ruled out when holding the highest card. For example, a player holding a 3 should never answer a bet with a pass, since by passing the player will lose, but by betting the player will win. Furthermore, when holding a 3, a player should always answer a pass with a bet, since in either case the player is going to win, but answering with a bet opens the possibility of the opponent betting again and thereby increasing the size of the win for the player holding the 3. Hence, when holding a 3,

player A should refrain from betting along line 1;

player B should refrain from betting along lines 1, 2, and 3.

Eliminating from consideration the above lines of betting, we see that player A now has 2 ×3 ×2 = 12 pure strategies and player B has 2 ×4 ×1 = 8 pure strategies. The number of pairs has therefore dropped to 96—a significant reduction. Not only do we eliminate these “bad” strategies from the mathematical model but also we assume that both players know that these bad strategies will not be used. That is, player A can assume that player B will play intelligently, and player B can assume the same of A. This knowledge then leads to further reductions. For example, when holding a 2, player A should refrain from betting along line 3. To reach this conclusion, we must carefully enumerate possibilities. Since player A holds the 2, player B holds either the 1 or the 3. But we’ve already determined what player B will do in both of those cases. Using this knowledge, it is not hard to see that player A would be unwise to bet along line 3. A similar analysis reveals that, when holding a 2, player B should refrain from lines 3 and 4. Therefore, player A now has only 2 ×2 ×2 = 8 pure strategies and player B has only 2 ×2 ×1 = 4 pure strategies.

At this point, no further reductions are possible. Computing the payoff matrix, we get

Solving the matrix game, we find that

and

These stochastic vectors can be summarized as simple statements of the optimal randomized strategies for the two players. Indeed, player A’s optimal strategy is as follows:

when holding 1, mix lines 1 and 3 in 5:1 proportion;

when holding 2, mix lines 1 and 2 in 1:1 proportion;

when holding 3, mix lines 2 and 3 in 1:1 proportion.

Similarly, player B’s optimal strategy can be described as

when holding 1, mix lines 1 and 3 in 2:1 proportion;

when holding 2, mix lines 1 and 2 in 2:1 proportion;

when holding 3, use line 4.

Note that it is optimal for player A to use line 3 when holding a 1 at least some of the time. Since line 3 says to bet, this bet is a bluff. Player B also bluffs sometimes, since betting line 3 is sometimes used when holding a 1. Clearly, the optimal strategies also exhibit some underbidding.

Exercises

11.1

Players A and B each hide a nickel or a dime. If the hidden coins match, player A gets both; if they don’t match, then B gets both. Find the optimal strategies. Which player has the advantage? Solve the problem for arbitrary denominations a and b.

11.2

Players A and B each pick a number between 1 and 100. The game is a draw if both players pick the same number. Otherwise, the player who picks the smaller number wins unless that smaller number is one less than the opponent’s number, in which case the opponent wins. Find the optimal strategy for this game.

11.3

We say that row rdominates row s if $$a_{rj} \geq a_{sj}$$ for all j = 1, 2, …, n. Similarly, column r is said to dominate column s if $$a_{ir} \geq a_{is}$$ for all i = 1, 2, …, m. Show that

(a)

If a row (say, r) dominates another row, then the row player has an optimal strategy y  ∗  in which $$y_{r}^{{\ast}} = 0$$.

(b)

If a column (say, s) is dominated by another column, then the column player has an optimal strategy x  ∗  in which $$x_{s}^{{\ast}} = 0$$.

Use these results to reduce the following payoff matrix to a 2 ×2 matrix:

11.4

Solve simplified poker assuming that antes are $2 and bets are $1.

11.5

Give necessary and sufficient conditions for the rth pure strategy of the row and the sth pure strategy of the column player to be simultaneously optimal.

11.6

Use the Minimax Theorem to show that

11.7

Bimatrix Games. Consider the following two-person game defined in terms of a pair of m ×n matrices A and B: if the row player selects row index i and the column player selects column index j, then the row player pays a ij dollars and the column player pays b ij dollars. Stochastic vectors x  ∗  and y  ∗  are said to form a Nash equilibrium if

The purpose of this exercise is to relate Nash equilibria to the problem of finding vectors x and y that satisfy

(11.5)

(vectors w and z can be thought of as being defined by the matrix equality). Problem (11.5) is called a linear complementarity problem.

(a)

Show that there is no loss in generality in assuming that A and B have all positive entries.

(b)

Assuming that A and B have all positive entries, show that, if $$({x}^{{\ast}},{y}^{{\ast}})$$ is a Nash equilibrium, then

solves the linear complementarity problem (11.5).

(c)

Show that, if (x′, y′) solves the linear complementarity problem (11.5), then

is a Nash equilibrium.

(An algorithm for solving the linear complementarity problem is developed in Exercise 18.7.)

11.8

The Game of Morra. Two players simultaneously throw out one or two fingers and call out their guess as to what the total sum of the outstretched fingers will be. If a player guesses right, but his opponent does not, he receives payment equal to his guess. In all other cases, it is a draw.

(a)

List the pure strategies for this game.

(b)

Write down the payoff matrix for this game.

(c)

Formulate the row player’s problem as a linear programming problem. (Hint: Recall that the row player’s problem is to minimize the maximum expected payout.)

(d)

What is the value of this game?

(e)

Find the optimal randomized strategy.

11.9

Heads I Win—Tails You Lose. In the classical coin-tossing game, player A tosses a fair coin. If it comes up heads player B pays player A $2 but if it comes up tails player A pays player B $2. As a two-person zero-sum game, this game is rather trivial since neither player has anything to decide (after agreeing to play the game). In fact, the matrix for this game is a 1 ×1 matrix with only a zero in it, which represents the expected payoff from player A to B.

Now consider the same game with the following twist. Player A is allowed to peek at the outcome and then decide either to stay in the game or to bow out. If player A bows out, then he automatically loses but only has to pay player B $1. Of course, player A must inform player B of his decision. If his decision is to stay in the game, then player B has the option either to stay in the game or not. If she decides to get out, then she loses $1 to player A. If both players stay in the game, then the rules are as in the classical game: heads means player A wins, tails means player B wins.

(a)

List the strategies for each player in this game. (Hint: Don’t forget that a strategy is something that a player has control over.)

(b)

Write down the payoff matrix.

(c)

A few of player A’s strategies are uniformly inferior to others. These strategies can be ruled out. Which of player A’s strategies can be ruled out?

(d)

Formulate the row player’s problem as a linear programming problem. (Hints: (1) Recall that the row player’s problem is to minimize the maximum expected payout. (2) Don’t include rows that you ruled out in the previous part.)

(e)

Find the optimal randomized strategy.

(f)

Discuss whether this game is interesting or not.

Notes

The Minimax Theorem was proved by von Neumann (1928). Important references include Gale et al. (1951), von Neumann and Morgenstern (1947), Karlin (1959), and Dresher (1961). Simplified poker was invented and analyzed by Kuhn (1950). Exercises 11.1 and 11.2 are borrowed from Chvátal (1983).

Bibliography

Chvátal, V. (1983). Linear programming. New York: Freeman.

Dresher, M. (1961). Games of strategy: Theory and application. Englewood Cliffs: Prentice-Hall.

Gale, D., Kuhn, H., and Tucker, A. (1951). Linear programming and the theory of games. In T. Koopmans (Ed.), Activity analysis of production and allocation (pp. 317–329). New York: Wiley.

Karlin, S. (1959). Mathematical methods and theory in games, programming, and economics (Vols. 1 and 2). Reading: Addison-Wesley.

Kuhn, H. (1950). A simplified two-person poker. Annals of Mathematics Studies, 24, 97–103.

von Neumann, J. (1928). Zur Theorie der Gesselschaftschpiele. Mathematische Annalen, 100, 295–320.Crossref

von Neumann, J., and Morgenstern, O. (1947). Theory of games and economic behavior (2nd ed.). Princeton: Princeton University Press.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_12

12. Regression

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In this chapter, we shall study an application of linear programming to an area of statistics called regression. As a specific example, we shall use size and iteration-count data collected from a standard suite of linear programming problems to derive a regression estimate of the number of iterations needed to solve problems of a given size.

1 Measures of Mediocrity

We begin our discussion with an example. Here are the midterm exam scores for a linear programming course:

Let m denote the number of exam scores (i.e., m = 9) and let b i , i = 1, 2, …, m, denote the actual scores (arranged in increasing order as above). The most naive measure of the “average” score is just the mean value, $$\bar{x}$$, defined by

This is an example of a statistic, which, by definition, is a function of a set of data. Statistics are computed so that one does not need to bother with reporting large tables of raw numbers. (Admittedly, the task of reporting the above list of nine exam scores is not very onerous, but this is just an example.) Now, suppose the professor in question did not report the scores but instead just gave summary statistics. Consider the student who got an 80 on the exam. This student surely didn’t feel great about this score but might have thought that at least it was better than average. However, as the raw data makes clear, this student really did worse than average¹ on the exam (the professor confesses that the exam was rather easy). In fact, out of the nine students, the one who got an 80 scored third from the bottom of the class. Furthermore, the student who scored worst on the exam did so badly that one might expect this student to drop the course, thereby making the 80 look even worse.

Any statistician would, of course, immediately suggest that we report the median score instead of the mean. The median score is, by definition, that score which is worse than half of the other scores and better than the other half. In other words, the median $$\hat{x}$$ is defined as

(Here and in various places in this chapter, we shall assume that m is odd so that certain formulas such as this one remain fairly simple.) Clearly, the 86 gives a more accurate indication of what the average score on the exam was.

There is a close connection between these statistical concepts and optimization. For example, the mean $$\bar{x}$$ minimizes, over all real numbers x, the sum of the squared deviations between the data points and x itself. That is,

To verify this claim, we let $$f(x) =\sum _{ i=1}^{m}{(x - b_{i})}^{2}$$, differentiate with respect to x, and set the derivative to zero to get

Solving this equation for the critical point² x, we see that

The fact that this critical point is a minimum rather than a maximum (or a saddle point) follows from the fact that f′(x) > 0 for all $$x \in \mathbb{R}$$.

The median $$\hat{x}$$ also enjoys a close connection with optimization. Indeed, it is the point that minimizes the sum of the absolute values of the difference between each data point and itself. That is,

To see that this is correct, we again use calculus. Let

This function is continuous, piecewise linear, and convex (see Figure 12.1). However, it is not differentiable at the data points. Nonetheless, we can look at its derivative at other points to see where it jumps across zero. The derivative, for $$x\not\in \{b_{1},b_{2},\ldots ,b_{m}\}$$, is

where

Hence, we see that the derivative at x is just the number of data points to the left of x minus the number of data points to the right. Clearly, this derivative jumps across zero at the median, implying that the median is the minimum.

Fig. 12.1

The objective function whose minimum occurs at the median.

In this chapter, we shall discuss certain generalizations of means and medians called regressions. At the end, we will consider a specific example that is of particular interest to us: the empirical average performance of the simplex method.

2 Multidimensional Measures: Regression Analysis

The analysis of the previous section can be recast as follows. Given a “random” observation b, we assume that it consists of two parts: a fixed, but unknown, part denoted by x and a random fluctuation about this fixed part, which we denote by ε. Hence,

Now, if we take several observations and index them as i = 1, 2, …, m, the b’s and the ε’s will vary, but x is assumed to be the same for all observations. Therefore, we can summarize the situation by writing

We now see that the mean is simply the value of x that minimizes the sum of the squares of the ε i ’is. Similarly, the median is the value of x that minimizes the sum of the absolute values of the ε i ’s.

Sometimes one wishes to do more than merely identify some sort of “average.” For example, a medical researcher might collect blood pressure data on thousands of patients with the aim of identifying how blood pressure depends on age, obesity (defined as weight over height), sex, etc. So associated with each observation b of a blood pressure are values of these control variables. Let’s denote by a ₁ the age of a person, a ₂ the obesity, a ₃ the sex, etc. Let n denote the number of different control variables being considered by the researcher. In (linear) regression analysis, we assume that the responseb depends linearly on the control variables. Hence, we assume that there are (unknown) numbers x j , j = 1, 2, …, n, such that

This equation is referred to as the regression model. Of course, the researcher collects data from thousands of patients, and so the data items, b and the a j ’s, must be indexed over these patients. That is,

If we let b denote the vector of observations, ε the vector of random fluctuations, and A the matrix whose ith row consists of the values of the control variables for the ith patient, then the regression model can be expressed in matrix notation as

(12.1)

In regression analysis, the goal is to find the vector x that best explains the observations b. Hence, we wish to pick values that minimize, in some sense, the vector ε’s. Just as for the mean and median, we can consider minimizing either the sum of the squares of the ε i ’s or the sum of the absolute values of the ε i ’s. There are even other possibilities. In the next two sections, we will discuss the range of possibilities and then give specifics for the two mentioned above.

3 L ²-Regression

There are several notions of the size of a vector. The most familiar one is the Euclidean length

This notion of length corresponds to our physical notion (at least when the dimension is low, such as 1, 2, or 3). However, one can use any power inside the sum as long as the corresponding root accompanies it on the outside of the sum. For 1 ≤ p < ∞, we get then the so-called L p -norm of a vector y

Other than p = 2, the second most important case is p = 1 (and the third most important case corresponds to the limit as p tends to infinity).

Measuring the size of ε in (12.1) using the L ²-norm, we arrive at the L ² -regression problem, which is to find $$\bar{x}$$ that attains the minimum L ²-norm for the difference between b and Ax. Of course, it is entirely equivalent to minimize the square of the L ²-norm, and so we get

Just as for the mean, there is an explicit formula for $$\bar{x}$$. To find it, we again rely on elementary calculus. Indeed, let

In this multidimensional setting, a critical point is defined as a point at which the derivative with respect to every variable vanishes. So if we denote a critical point by $$\bar{x}$$, we see that it must satisfy the following equations:

Simplifying these equations, we get

In matrix notation, these equations can be summarized as follows:

In other words, assuming that A T A is invertible, we get

(12.2)

This is the formula for L ²-regression. It is also commonly called least squares regression. In Section 12.6, we will use this formula to solve a specific regression problem.

Example. The simplest and most common regression model arises when one wishes to describe a response variable b as a linear function of a single input variable a. In this case, the model is

The unknowns here are the slope x ₁ and the intercept x ₂. Figure 12.2 shows a plot of three pairs (a, b) through which we want to draw the “best” straight line. At first glance, this model does not seem to fit the regression paradigm, since regression models (as we’ve defined them) do not involve a term for a nonzero intercept. But the model here can be made to fit by introducing a new control variable, say, a ₂, which is always set to 1. While we’re at it, let’s change our notation for a to a ₁ so that the model can now be written as

The three data points can then be summarized in matrix notation as

For this problem,

and

Hence,

Fig. 12.2

Three data points for a linear regression.

4 L ¹-Regression

Just as the median gives a more robust estimate of the “average value” of a collection of numbers than the mean, L ¹-regression is less sensitive to outliers than least squares regression is. It is defined by minimizing the L ¹-norm of the deviation vector in (12.1). That is, the problem is to find $$\hat{x}$$ as follows:

Unlike for least squares regression, there is no explicit formula for the solution to the L ¹-regression problem. However, the problem can be reformulated as a linear programming problem. Indeed, it is easy to see that the L ¹-regression problem,

can be rewritten as

which is equivalent to the following linear programming problem:

(12.3)

Hence, to solve the L ¹-regression problem, it suffices to solve this linear programming problem. In the next section, we shall present an alternative algorithm for computing the solution to an L ¹-regression problem.

Example. Returning to the example of the last section, the L ¹-regression problem is solved by finding the optimal solution to the following linear programming problem:

The solution to this linear programming problem is

which clearly indicates that the point (2, 2. 5) is viewed by the L ¹-regression as an outlier, since the regression line passes exactly through the other two points.

5 Iteratively Reweighted Least Squares

Even though calculus cannot be used to obtain an explicit formula for the solution to the L ¹-regression problem, it can be used to obtain an iterative procedure that, when properly initialized, converges to the solution of the L ¹-regression problem. The resulting iterative process is called iteratively reweighted least squares. In this section, we briefly discuss this method. We start by considering the objective function for L ¹-regression:

Differentiating this objective function is a problem, since it involves absolute values. However, the absolute value function

is differentiable everywhere except at one point: z = 0. Furthermore, we can use the following simple formula for the derivative, where it exists:

Using this formula to differentiate f with respect to each variable, and setting the derivatives to zero, we get the following equations for critical points:

(12.4)

If we introduce the following shorthand notation for the deviations,

we see that we can rewrite (12.4) as

Now, if we let E x denote the diagonal matrix containing the vector ε(x) on the diagonal, we can write these equations in matrix notation as follows:

This equation can’t be solved for x as we were able to do in L ²-regression because of the dependence of the diagonal matrix on x. But let us rearrange this system of equations by multiplying both sides by the inverse of $${A}^{T}E_{x}^{-1}A$$. The result is

This formula suggests an iterative scheme that hopefully converges to a solution. Indeed, we start by initializing x ⁰ arbitrarily and then use the above formula to successively compute new approximations. If we let x k denote the approximation at the kth iteration, then the update formula can be expressed as

Assuming only that the matrix inverse exists at every iteration, one can show that this iteration scheme converges to a solution to the L ¹-regression problem.

6 An Example: How Fast Is the Simplex Method?

In Chapter 4, we discussed the worst-case behavior of the simplex method and studied the Klee–Minty problem that achieves the worst case. We also discussed the importance of empirical studies of algorithm performance. In this section, we shall introduce a model that allows us to summarize the results of these empirical studies.

We wish to relate the number of simplex iterations T required to solve a linear programming problem to the number of constraints m and/or the number of variables n in the problem (or some combination of the two). As any statistician will report, the first step is to introduce an appropriate model.³ Hence, we begin by asking: how many iterations, on average, do we expect the simplex method to take if the problem has m constraints and n variables? To propose an answer to this question, consider the initial dictionary associated with a given problem. This dictionary involves m values, $$x_{\mathcal{B}}^{{\ast}}$$, for the primal basic variables, and n values, $$y_{\mathcal{N}}^{{\ast}}$$, for the dual nonbasic variables. We would like each of these m + n variables to have nonnegative values, since that would indicate optimality. If we assume that the initial dictionary is nondegenerate, then one would expect on the average that $$(m + n)/2$$ of the values would be positive and the remaining $$(m + n)/2$$ values would be negative.

Now let’s look at the dynamics of the simplex method. Each iteration focuses on exactly one of the negative values. Suppose, for the sake of discussion, that the negative value corresponds to a dual nonbasic variable, that is, one of the coefficients in the objective row of the dictionary. Then the simplex method selects the corresponding primal nonbasic variable to enter the basis, and a leaving variable is chosen by a ratio test. After the pivot, the variable that exited now appears as a nonbasic variable in the same position that the entering variable held before. Furthermore, the coefficient on this variable is guaranteed to be positive (since we’ve assumed nondegeneracy). Hence, the effect of one pivot of the simplex method is to correct the sign of one of the negative values from the list of m + n values of interest. Of course, the pivot also affects all the other values, but there seems no reason to assume that the situation relative to them will have any tendency to get better or worse, on the average. Therefore, we can think of the simplex method as statistically reducing the number of negative values by one at each iteration.

Since we expect on the average that an initial dictionary will have $$(m + n)/2$$ negative values, it follows that the simplex method should take $$(m + n)/2$$ iterations, on average. Of course, these expectations are predicated on the assumption that degenerate dictionaries don’t arise. As we saw in Section 7.2, the self-dual simplex method initialized with random perturbations will, with probability one, never encounter a degenerate dictionary. Hence, we hypothesize that this variant of the simplex method will, on average, take $$(m + n)/2$$ iterations. It is important to note the main point of our hypothesis; namely, that the number of iterations is linear in m + n as opposed, say, to quadratic or cubic.

We can test our hypothesis by first supposing that T can be approximated by a function of the form

for a pair of real numbers α and β. Our goal then is to find the value for these parameters that best fits the data obtained from a set of empirical observations. (We’ve written the leading constant as 2 α simply for symmetry with the other factor—there is no fundamental need to do this.) This multiplicative representation of the number of iterations can be converted into an additive (in α and β) representation by taking logarithms. Introducing an ε to represent the difference between the model’s prediction and the true number of iterations, we see that the model can be written as

Now, suppose that several observations are made. Using subscripts to distinguish the various observations, we get the following equations:

If we let b denote the vector on the left, A the matrix on the right, x the vector multiplied by A, and ε the vector of deviations, then the model can be expressed as

where A and b are given. As we’ve seen, this is just a regression model, which we can solve as an L ¹-regression or as an L ²-regression.

Given real data, we shall solve this model both ways. Table 12.1 shows specific data obtained by running the self-dual simplex method described in Chapter 7 (with randomized initial perturbations) against most of the problems in a standard suite of test problems (called the netlib suite Gay 1985). Some problems were too big to run on the workstation used for this experiment, and others were formulated with free variables that the code was not equipped to handle.

Table 12.1

Number of iterations for the self-dual simplex method.

Name

m

n

iters

Name

m

n

iters

25fv47

777

1,545

5,089

nesm

646

2,740

5,829

80bau3b

2,021

9,195

10,514

recipe

74

136

80

adlittle

53

96

141

sc105

104

103

92

afiro

25

32

16

sc205

203

202

191

agg2

481

301

204

sc50a

49

48

46

agg3

481

301

193

sc50b

48

48

53

bandm

224

379

1,139

scagr25

347

499

1,336

beaconfd

111

172

113

scagr7

95

139

339

blend

72

83

117

scfxm1

282

439

531

bnl1

564

1,113

2,580

scfxm2

564

878

1,197

bnl2

1,874

3,134

6,381

scfxm3

846

1,317

1,886

boeing1

298

373

619

scorpion

292

331

411

boeing2

125

143

168

scrs8

447

1,131

783

bore3d

138

188

227

scsd1

77

760

172

brandy

123

205

585

scsd6

147

1,350

494

czprob

689

2,770

2,635

scsd8

397

2,750

1,548

d6cube

403

6,183

5,883

sctap1

284

480

643

degen2

444

534

1,421

sctap2

1,033

1,880

1,037

degen3

1,503

1,818

6,398

sctap3

1,408

2,480

1,339

e226

162

260

598

seba

449

896

766

etamacro

334

542

1,580

share1b

107

217

404

fffff800

476

817

1,029

share2b

93

79

189

finnis

398

541

680

shell

487

1,476

1,155

fit1d

24

1,026

925

ship04l

317

1,915

597

fit1p

627

1,677

15,284

ship04s

241

1,291

560

forplan

133

415

576

ship08l

520

3,149

1,091

ganges

1,121

1,493

2,716

ship08s

326

1,632

897

greenbea

1,948

4,131

21,476

ship12l

687

4,224

1,654

grow15

300

645

681

ship12s

417

1,996

1,360

grow22

440

946

999

sierra

1,212

2,016

793

grow7

140

301

322

standata

301

1,038

74

israel

163

142

209

standmps

409

1,038

295

kb2

43

41

63

stocfor1

98

100

81

lotfi

134

300

242

stocfor2

2,129

2,015

2,127

maros

680

1,062

2,998

Using (12.2) to solve the problem as an L ²-regression, we get

Or, in other words,

This is amazingly close to our hypothesized formula, $$(m + n)/2$$. Figure 12.3 shows a log–log plot of T vs. m + n with the L ²-regression line drawn through it. It is clear from this graph that a straight line (in the log–log plot) is a good model for fitting this data.

Fig. 12.3

A log–log plot of T vs. m + n and the L ¹ and L ² regression lines.

Using (12.3) to solving the problem, we get

In other words,

The fact that this regression formula agrees closely with the L ²-regression indicates that the data set contains no outliers. In Section 12.6.1, we will consider randomly generated problems and see at least one example where the L ¹ and L ² regression lines differ significantly.

6.1 Random Problems

Now, let’s consider random problems generated in a manner similar to the way we did it back in Chapter 4. We do, however, introduce some changes. First of all, the problems in Chapter 4 were generated in such a manner as to guarantee primal feasibility but dual feasibility was left to chance—that is, many (half) of the problems were unbounded. The problems we wish to consider now will be assumed to have optimal solutions (real-world problems are often, but not always, known to have an optimal solution because of the underlying physical model and therefore primal or dual infeasibility is often an indicator of data and/or modeling errors). To guarantee the existence of an optimal solution, we generate random optimal primal and dual solutions and associated random optimal slack/surplus variables. Here’s the matlab code for that:

x = round(sigma*rand(n,1)).*(rand(n,1)>0.5);

y = round(sigma*rand(1,m)).*(rand(1,m)>0.5);

z = round(sigma*rand(1,n)).*(rand(1,n)>0.5);

w = round(sigma*rand(m,1)).*(rand(m,1)>0.5);

(as in Chapter 4, sigma is a constant initialized to 10). We then define A, b, and c as problem data consistent with these optimal solutions:

A = round(sigma*(randn(m,n))).*(rand(m,n)>0.5);

b = A*x + w;

c = y*A - z;

Note that we have made one other key change from before—we have randomly forced about half of the optimal values of the variables and about half of the constraint matrix coefficients to be zero. This change makes the problems slightly more realistic as real-world problems often have much sparsity.

Next, we need to initialize a right-hand side perturbation and an objective function perturbation:

b0 = rand(m,1);

c0 = -rand(1,n);

There are just three relatively simple changes to the code defining the simplex method itself to convert it from a primal-simplex algorithm to the parametric self-dual method. The first is to change the line of code that checks if the problem has been solved. Before, we only needed to check if all of the objective coefficients had become negative (dual feasibility) because primal feasibility was built in. Now, we have to check both primal and dual feasibility:

while max(c) > eps || min(b) < -eps,

Secondly, the choice of enter/leaving variables must be updated as it is now based on minimizing the perturbation parameter:

[mu_col, col] = max( (-c./c0).*(c0<-eps));

[mu_row, row] = max( (-b./b0).*(b0> eps));

if mu_col >= mu_row,

mu = mu_col;

Acol = A(:,col);

[t, row] = max(-Acol./(b+mu*b0));

else

mu = mu_row;

Arow = A(row,:);

[s, col] = max(-Arow./(c+mu*c0));

end

Finally, as part of every pivot we have to update b0 and c0:

brow = b0(row);

b0 = b0 - brow*Acol/a;

b0(row) = -brow/a;

ccol = c0(col);

c0 = c0 - ccol*Arow/a;

c0(col) = ccol/a;

The code was run 1,000 times. Figure 12.4 shows the number of pivots plotted against the sum m + n. Just as we saw with the primal simplex method in Chapter 4, m + n does not seem to be a good measure of problem size as many problems of a given size solve much more quickly than the more typical cases. Hence, there are a number of “outliers.” Overlayed on the scatter plot are the L ¹ and L ² regression lines. While neither regression line follows what appears to be an upper line of points that seems to dominate the results, the L ¹ is closer to that than is the L ² line.

Fig. 12.4

The parametric self-dual simplex method was used to solve 1,000 problems known to have an optimal solution. Shown here is a log-log plot showing the number of pivots required to reach optimality plotted against m + n. Also shown are the L ¹ and L ² regression lines.

The result of the L ¹-regression is:

The result of the L ²-regression is:

Finally, as in Chapter 4, min(m, n) is a better measure of problem size for these randomly generated problems. Figure 12.5 shows the same data plotted against min(m, n).

Fig. 12.5

The parametric self-dual simplex method was used to solve 1,000 problems known to have an optimal solution. Shown here is a log-log plot showing the number of pivots required to reach optimality plotted against min(m, n). In this case, the L ¹ and L ² regression lines are almost exactly on top of each other.

In this case, both regression lines are about the same:

Exercises

12.1

Find the L ²-regression line for the data shown in Figure 12.6.

Fig. 12.6

Four data points for a linear regression.

12.2

Find the L ¹-regression line for the data shown in Figure 12.6.

12.3

Midrange. Given a sorted set of real numbers, $$\{b_{1},b_{2},\ldots ,b_{m}\}$$, show that the midrange, $$\tilde{x} = (b_{1} + b_{m})/2$$, minimizes the maximum deviation from the set of observations. That is,

12.4

Centroid. Given a set of points $$\{b_{1},b_{2},\ldots ,b_{m}\}$$ on the plane $${\mathbb{R}}^{2}$$, show that the centroid

minimizes the sum of the squares of the distance to each point in the set. That is, $$\bar{x}$$ solves the following optimization problem:

Note: Each data point b i is a vector in $${\mathbb{R}}^{2}$$ whose components are denoted, say, by b i1 and b i2 , and, as usual, the subscript 2 on the norm denotes the Euclidean norm. Hence,

12.5

Facility Location. A common problem is to determine where to locate a facility so that the distance from its customers is minimized. That is, given a set of points $$\{b_{1},b_{2},\ldots ,b_{m}\}$$ on the plane $${\mathbb{R}}^{2}$$, the problem is to find $$\hat{x} = (\hat{x}_{1},\hat{x}_{2})$$ that solves the following optimization problem:

As for L ¹-regression, there is no explicit formula for $$\hat{x}$$, but an iterative scheme can be derived along the same lines as in Section 12.5. Derive an explicit formula for this iteration scheme.

12.6

A Simple Steiner Tree. Suppose there are only three customers in the facility location problem of the previous exercise. Suppose that the triangle formed by b ₁, b ₂, and b ₃ has no angles greater than 120 ∘ . Show that the solution $$\hat{x}$$ to the facility location problem is the unique point in the triangle from whose perspective the three customers are 120 ∘  apart from each other. What is the solution if one of the angles, say, at vertex b ₁, is more than 120 ∘ ?

12.7

Sales Force Planning. A distributor of office equipment finds that the business has seasonal peaks and valleys. The company uses two types of sales persons: (a) regular employees who are employed year-round and cost the company $17.50/h (fully loaded for benefits and taxes) and (b) temporary employees supplied by an outside agency at a cost of $25/h. Projections for the number of hours of labor by month for the following year are shown in Table 12.2. Let a i denote the number of hours of labor needed for month i and let x denote the number of hours per month of labor that will be handled by regular employees. To minimize total labor costs, one needs to solve the following optimization problem:

(a)

Show how to reformulate this problem as a linear programming problem.

(b)

Solve the problem for the specific data given above.

(c)

Use calculus to find a formula giving the optimal value for x.

12.8

Acceleration Due to Gravity. The law of gravity from classical physics says that an object dropped from a tall building will, in the absence of air resistance, have a constant rate of acceleration g so that the height x, as a function of time t, is given by

Unfortunately, the effects of air resistance cannot be ignored. To include them, we assume that the object experiences a retarding force that is directly proportional to its speed. Letting v(t) denote the velocity of the object at time t, the equations that describe the motion are then given by

(f is the unknown constant of proportionality from the air resistance). These equations can be solved explicitly for x as a function of t:

It is clear from the equation for the velocity that the terminal velocity is g ∕ f. It would be nice to be able to compute g by measuring this velocity, but this is not possible, since the terminal velocity involves both f and g. However, we can use the formula for x(t) to get a two-parameter model from which we can compute both f and g. Indeed, if we assume that all measurements are taken after terminal velocity has been “reached” (i.e., when e  − ft is much smaller than 1), then we can write a simple linear expression relating position to time:

Now, in our experiments we shall set values of x (corresponding to specific positions below the drop point) and measure the time at which the object passes these positions. Since we prefer to write regression models with the observed variable expressed as a linear function of the control variables, let us rearrange the above expression so that t appears as a function of x:

Using this regression model and the data shown in Table 12.3, do an L ²-regression to compute estimates for 1 ∕ f and $$-f/g$$. From these estimates derive an estimate for g. If you have access to linear programming software, solve the problem using an L ¹-regression and compare your answers.

12.9

Iteratively Reweighted Least Squares. Show that the sequence of iterates in the iteratively reweighted least squares algorithm produces a monotonically decreasing sequence of objective function values by filling in the details in the following outline. First, recall that the objective function for L ¹-regression is given by

where

Also, the function that defines the iterative scheme is given by

where E x denotes the diagonal matrix with the vector ε(x) on its diagonal. Our aim is to show that

In order to prove this inequality, let

(a)

Use calculus to show that, for each x, T(x) is a global minimum of g x .

(b)

Show that g x (x) = f(x).

(c)

By writing

and then substituting the right-hand expression into the definition of g x (T(x)), show that

(d)

Combine the three steps above to finish the proof.

12.10

In our study of means and medians, we showed that the median of a collection of numbers, $$b_{1},b_{2},\ldots ,b_{n}$$, is the number $$\hat{x}$$ that minimizes $$\sum _{j}\vert x - b_{j}\vert $$. Let μ be a real parameter.

(a)

Give a statistical interpretation to the following optimization problem:

Hint: the special cases $$\mu = 0,\pm 1/2,\pm 1$$ might help clarify the general situation.

(b)

Express the above problem as a linear programming problem.

(c)

The parametric simplex method can be used to solve families of linear programming problems indexed by a parameter μ (such as we have here). Starting at μ = ∞ and proceeding to $$\mu = -\infty $$ one solves all of the linear programs with just a finite number of pivots. Use the parametric simplex method to solve the problems of the previous part in the case where n = 4 and b ₁ = 1, b ₂ = 2, b ₃ = 4, and b ₄ = 8.

(d)

Now consider the general case. Write down the dictionary that appears in the k-th iteration and show by induction that it is correct.

12.11

Show that the L ∞ -norm is just the maximum of the absolute values. That is,

Table 12.2

Projected labor hours by month.

Jan

390

May

310

Sep

550

Feb

420

Jun

590

Oct

360

Mar

340

Jul

340

Nov

420

Apr

320

Aug

580

Dec

600

Table 12.3

Time at which a falling object passes certain points.

Obs.

Position

Time

number

(m)

(s)

1

 − 10

3.72

2

 − 20

7.06

3

 − 30

10.46

4

 − 10

3.71

5

 − 20

7.00

6

 − 30

10.48

7

 − 10

3.67

8

 − 20

7.08

9

 − 30

10.33

Notes

Gonin and Money (1989) and Dodge (1987) are two references on regression that include discussion of both L ² and L ¹ regression. The standard reference on L ¹ regression is Bloomfield and Steiger (1983).

Several researchers, including Smale (1983), Borgwardt (1982, 1987a), Adler and Megiddo (1985), and Todd (1986), have studied the average number of iterations of the simplex method as a function of m and/or n. The model discussed in this chapter is similar to the sign-invariant model introduced by Adler and Berenguer (1981).

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Footnotes

1

“Average” is usually taken as synonymous with “mean” but in this section we shall use it in an imprecise sense, employing other technically defined terms for specific meanings.

2

Recall from calculus that a critical point is any point at which the derivative vanishes or fails to exist.

3

In the social sciences, a fundamental difficulty is the lack of specific arguments validating the appropriateness of the models commonly introduced.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_13

13. Financial Applications

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In this chapter, we shall study some applications of linear programming to problems in quantitative finance.

1 Portfolio Selection

Every investor, from the individual to the professional fund manager, must decide on an appropriate mix of assets to include in his or her investment portfolio. Given a collection of potential investments (indexed, say, from 1 to n), let R j denote the return in the next time period on investment j, j = 1, …, n. In general, R j is a random variable, although some investments may be essentially deterministic.

A portfolio is determined by specifying what fraction of one’s assets to put into each investment. That is, a portfolio is a collection of nonnegative numbers x j , j = 1, …, n, that sum to one. The return (on each dollar) one would obtain using a given portfolio is given by

The reward associated with such a portfolio is defined as the expected return¹:

If reward were the only issue, then the problem would be trivial: simply put everything in the investment with the highest expected return. But unfortunately, investments with high reward typically also carry a high level of risk. That is, even though they are expected to do very well in the long run, they also tend to be erratic in the short term. There are many ways to define risk, some better than others. We will define the risk associated with an investment (or, for that matter, a portfolio of investments) to be the mean absolute deviation from the mean (MAD):

where $$\tilde{R}_{j} = R_{j} - \mathbb{E}R_{j}$$. One would like to maximize the reward while at the same time not incur excessive risk. Whenever confronted with two (or more) competing objectives, it is necessary to consider a spectrum of possible optimal solutions as one moves from putting most weight on one objective to the other. In our portfolio selection problem, we form a linear combination of the reward and the risk (parametrized here by μ) and maximize that:

(13.1)

Here, μ is a positive parameter that represents the importance of risk relative to reward: high values of μ tend to maximize reward regardless of risk, whereas low values attempt to minimize risk.

It is important to note that by diversifying (that is, not putting everything into one investment), it might be possible to reduce the risk without reducing the reward. To see how this can happen, consider a hypothetical situation involving two investments A and B. Each year, investment A either goes up 30% or goes down 10%, but unfortunately, the ups and downs are unpredictable (that is, each year is independent of the previous years and is an up year with probability 1 ∕ 2). Investment B is also highly volatile. In fact, in any year in which A goes up 30%, investment B goes down 10%, and in the years in which A goes down 10%, B goes up 30%. Clearly, by putting half of our portfolio into A and half into B, we can create a portfolio that goes up 10% every year without fail. The act of identifying investments that are negatively correlated with each other (such as A and B) and dividing the portfolio among these investments is called hedging. Unfortunately, it is fairly difficult to find pairs of investments with strong negative correlations. But such negative correlations do occur. Generally speaking, they can be expected to occur when the fortunes of both A and B depend on a common underlying factor. For example, a hot, rainless summer is good for energy but bad for agriculture.

Solving problem (13.1) requires knowledge of the joint distribution of the R j ’s. However, this distribution is not known theoretically but instead must be estimated by looking at historical data. For example, Table 13.1 shows monthly returns over a recent 2-year period for one bond fund (3-year Treasury Bonds) and eight different sector index funds: Materials (XLB), Energy (XLE), Financial (XLF), Industrial (XLI), Technology (XLK), Staples (XLP), Utilities (XLU), and Healthcare (XLV). Let R j (t) denote the return on investment j over T monthly time periods as shown in Table 13.1. One way to estimate the mean $$\mathbb{E}R_{j}$$ is simply to take the average of the historical returns:

Table 13.1

Monthly returns per dollar for each of nine investments over 2 years. That is, $1 invested in the energy sector fund XLE on April 1, 2007, was worth $1. 068 on April 30, 2007.

Year-

SHY

XLB

XLE

XLF

XLI

XLK

XLP

XLU

XLV

Month

Bonds

Materials

Energy

Financial

Indust.

Tech.

Staples

Util.

Health

2007-04

1.000

1.044

1.068

1.016

1.035

1.032

1.004

0.987

1.014

2007-03

1.003

1.015

1.051

1.039

1.046

1.047

1.028

1.049

1.073

2007-02

1.005

1.024

1.062

0.994

1.008

1.010

1.021

1.036

1.002

2007-01

1.007

1.027

0.980

0.971

0.989

0.973

0.985

1.053

0.977

2006-12

1.002

1.040

0.991

1.009

1.021

1.020

1.020

0.996

1.030

2006-11

1.001

0.995

0.969

1.030

0.997

0.989

1.020

0.999

1.007

2006-10

1.005

1.044

1.086

1.007

1.024

1.028

0.991

1.026

0.999

2006-09

1.004

1.060

1.043

1.023

1.028

1.040

1.018

1.053

1.003

2006-08

1.004

1.000

0.963

1.040

1.038

1.040

0.999

0.985

1.015

2006-07

1.008

1.030

0.949

1.012

1.011

1.070

1.039

1.028

1.029

2006-06

1.007

0.963

1.034

1.023

0.943

0.974

1.016

1.048

1.055

2006-05

1.002

1.005

1.022

0.995

0.999

0.995

1.018

1.023

1.000

2006-04

1.002

0.960

0.972

0.962

0.983

0.935

1.002

1.016

0.979

2006-03

1.002

1.035

1.050

1.043

1.021

0.987

1.010

1.016

0.969

2006-02

1.002

1.047

1.042

1.003

1.044

1.023

1.008

0.954

0.987

2006-01

1.000

0.978

0.908

1.021

1.031

1.002

1.008

1.013

1.012

2005-12

1.002

1.048

1.146

1.009

1.003

1.034

1.002

1.024

1.013

2005-11

1.004

1.029

1.018

1.000

1.005

0.969

1.001

1.009

1.035

2005-10

1.004

1.076

1.015

1.048

1.058

1.063

1.009

0.999

1.012

2005-09

0.999

1.002

0.909

1.030

0.986

0.977

0.996

0.936

0.969

2005-08

0.997

1.008

1.063

1.009

1.017

1.002

1.014

1.042

0.995

2005-07

1.007

0.958

1.064

0.983

0.976

0.991

0.983

1.006

0.996

2005-06

0.996

1.056

1.071

1.016

1.038

1.057

1.032

1.023

1.023

2005-05

1.002

0.980

1.070

1.012

0.974

0.987

0.981

1.059

0.994

1.1 Reduction to a Linear Programming Problem

As formulated, the problem in (13.1) is not a linear programming problem. We use the same trick we used in the previous chapter to replace each absolute value with a new variable and then impose inequality constraints that ensure that the new variable will indeed be the appropriate absolute value once an optimal value to the problem has been obtained. But first, let us rewrite (13.1) with the expected value operation replaced by a simple averaging over the given historical data:

(13.2)

where

denotes the expected reward for asset j. Now, replace $$\left \vert \sum _{j}x_{j}(R_{j}(t) - r_{j})\right \vert $$ with a new variable y t and rewrite the optimization problem as

(13.3)

As we’ve seen in other contexts before, at optimality one of the two inequalities involving y t must actually be an equality because if both inequalities were strict then it would be possible to further increase the objective function by reducing y t .

1.2 Solution via Parametric Simplex Method

The problem formulation given by (13.3) is a linear program that can be solved for any particular value of μ using the methods described in previous chapters. However, we can do much better than this. The problem is a parametric linear programming problem, where the parameter is the risk aversion parameter μ. If we can give a value of μ for which a basic optimal solution is obvious, then we can start from this basic solution and use the parametric simplex method to find the optimal solution associated with each and every value of μ. It is easy to see that for μ larger than some threshold, the optimal solution is to put all of our portfolio into a single asset, the one with the highest expected reward r j . Let j  ∗  denote this highest reward asset:

We need to write (13.3) in dictionary form. To this end, let us introduce slack variables w t  +  and w t  − :

We have 3T + n nonnegative variables and 2T + 1 equality constraints. Hence, we need to find 2T + 1 basic variables and $$T + n - 1$$ nonbasic variables. Since we know the optimal values for each of the allocation variables, $$x_{{j}^{{\ast}}} = 1$$ and the rest of the x j ’s vanish, it is straightforward to figure out the values of the other variables as well. We can then simply declare any variable that is positive to be basic and declare the rest to be nonbasic. With this prescription, the variable $$x_{{j}^{{\ast}}}$$ must be basic. The remaining x j ’s are nonbasic. Similarly, all of the y t ’s are nonzero and hence basic. For each t, either w t  −  or w t  +  is basic and the other is nonbasic. To say which is which, we need to introduce some additional notation. Let

Then it is easy to check that w t  −  is basic if $$D_{t{j}^{{\ast}}} > 0$$ and w t  +  is basic if $$D_{t{j}^{{\ast}}} < 0$$ (the unlikely case where $$D_{t{j}^{{\ast}}} = 0$$ can be decided arbitrarily). Let

and let

It’s tedious, but here’s the optimal dictionary:

We can now check that, for large μ, this dictionary is optimal. Indeed, the objective coefficients on the w t  −  and w t  +  variables in the first row of the objective function are negative. The coefficients on the x j ’s in the first row can be positive or negative but for μ sufficiently large, the negative coefficients on the x j ’s in the second row dominate and make all coefficients negative after considering both rows. Similarly, the fact that all of the basic variables are positive follows immediately from the definitions of T  +  and T  − .

A few simple inequalities determine the μ-threshold above which the given dictionary is optimal. The parametric simplex method can then be used to systematically reduce μ to zero. Along the way, each dictionary encountered corresponds to an optimal solution for some range of μ values. Hence, in one pass we have solved the portfolio selection problem for every investor from the bravest to the most cautious. Figure 13.1 shows all of the optimal portfolios. The set of all risk–reward profiles that are possible is shown in Figure 13.2.  The lower-right boundary of this set is the so-called efficient frontier. Any portfolio that produces a risk–reward combination that does not lie on the efficient frontier can be improved either by increasing its mean reward without changing the risk or by decreasing the risk without changing the mean reward. Hence, one should only invest in portfolios that lie on the efficient frontier.

FIGURE 13.1.

Optimal portfolios as a function of risk parameter μ.

Fig. 13.2

The efficient frontier.

2 Option Pricing

Option pricing is one of the fundamental problems of quantitative finance. In this section we will describe briefly what an option is and formulate upper and lower bounds on the price as a linear programming problem.

An option is a derivative security, which means that it is derived from a simpler security such as a stock. There are many types of options, some quite exotic. For the purposes of this book, I will only describe the simplest type of option, the call option. A call option is a contract between two parties in which one party, the buyer, is given the option to buy from the other party a particular stock at a particular price at a particular time some weeks or months in the future. For example, on June 1st, 2007, Apple Computer stock was selling for $121 per share. On this date, it was possible to buy an option allowing one to purchase Apple stock for $130 (the so-called strike price) a share 10 weeks in the future (the expiration date). The seller was offering this contract for a price of $3. 20. Where does this price come from? The simple answer is that it is determined solely by the marketplace, since option contract themselves can be bought and sold up until their expiration date. But, as technical folks, we seek an analytical formula that tells us what a fair price ought to be. This we can do.

To explain how to price the option, we need to think a little bit more about the value of the option on the date of expiration. If Apple stock does well over the next 10 weeks and ends up at $140 per share, then on the day of expiration I can exercise the contract and buy the stock for $130. I can then immediately sell the stock for $140 and pocket the $10 difference. Of course, I paid $3. 20 for the right to do this. Hence, my net profit is $6. 80. Now, suppose instead of rising to $140 per share, the stock only climbs to $132 per share. In this case, I will still want to exercise the option because I can pocket a $2 difference. But, after subtracting the cost of the option, I’ve actually lost a modest $1. 20 per share. Finally, suppose that the stock only goes up to $125 per share. In this case, I will let the option expire without exercising it. I will have lost only the $3. 20 that I originally paid for the option. Finally, consider the case where in the intervening 10 weeks some really bad news surfaces that drives Apple stock down to $100 per share. Had I actually bought Apple stock, I would now be out $21 per share, which could be a substantial amount of money if I had bought lots of shares. But, by buying the option, I’m only out $3. 20 per share. This is the attraction of call options. They allow an investor who is optimistic about the economy (or a particular company) to take a chance without risking much on the down side. Figure 13.3 shows a plot of the net profit per share as a function of the share price at expiration.

Fig. 13.3

A graph of the value of the option at expiration as a function of stock price. In this example, the strike price is $130.

Let s ₀ denote the (known) current stock price and let S ₁ denote the (not yet known, i.e., random) stock price at expiration. A key feature of options is that their value at the expiration date is given by a specific function h(S ₁) of the stock price at expiration. For the specific call option discussed above, the function h(S ₁) is the “hockey-stick” shaped function shown in Figure 13.3. If we think we know the distribution of the random variable S ₁, then we could compute its expected value and, if we ignore the discounting for inflation, we could use this to price the option:

Unfortunately, we generally don’t know the distribution of S ₁.

We can, however, make some indirect inferences based on “market wisdom” that constrain the possible values for p and thereby implicitly tell us something about the distribution of S ₁. Specifically, let us imagine that there are already a number of options being traded in the market that are based on the same underlying stock and have the same expiration date. Let us suppose that there are already n options being traded in the market with known prices. That is, there are specific functions h j (S ₁), j = 1, 2, …, n, for which there are already known prices p j . One can think of the underlying stock itself as the simplest possible option. Since the stock is traded, it too provides some information about the future. We assume that this trivial asset is the j = 1 option in the collection of known priced options. For this option, we have $$h_{1}(S_{1}) = S_{1}$$ and p ₁ = s ₀. To these n options, we add one more: cash. One dollar today will be worth one dollar on the expiration date (again, we are ignoring here the time value of money). In a sense, this is also and option.

The problem we wish to consider is how to price a new option whose payout function we denote by g(S ₁). Consider building a portfolio of the available options consisting of x ₀ “shares” of dollars, x ₁ shares of the underlying stock, and x j shares of option j (j = 2, …, n). Today, this portfolio costs

At the expiration date, the portolio’s value will be

Suppose that no matter how S ₁ turns out, the value of the new option dominates that of the portfolio:

Then, it must be the case that the price p of the new option today must also dominate the cost of this portfolio:

This is called a no-arbitrage condition. This no-arbitrage condition implies a lower bound $$\underline{p}$$ on the price of the new option, which we can maximize:

This problem actually has an infinite number of constraints because the inequality must hold no matter what value S ₁ takes on. It can be made into a linear programming problem by introducing a finite set of possible values, say $$s_{1}(1),s_{1}(2),\ldots,s_{1}(m)$$. The resulting linear programming problem can thus be written as

(13.4)

In a completely analogous manner we can find a tight upper bound $$\bar{p}$$ for p by solving a minimization problem:

(13.5)

The dual problem associated with (13.4) is

Note that the first and last constraints tell us that the y i ’s are a system of probabilities. Given this interpretation of the y i ’s as probabilities, the expression

is just an expected value of the random variable S ₁ computed using these probabilities. So, the constraint $$\sum _{i}s_{1}(i)y_{i} = s_{0}$$ means that the expected stock price at the end of the time period must match the current stock price, when computed with the y i probabilities. For this reason, we call these probabilities risk neutral. Similarly, the constraints $$\sum _{i}h_{j}(s_{1}(i))y_{i} = p_{j}$$, j = 2, …, n, tell us that each of the options must also be priced in such a way that the expected future price matches the current market price.

Exercises

13.1

Find every portfolio on the efficient frontier using the most recent 6 months of data for the Bond (SHY), Materials (XLB), Energy (XLE), and Finacial (XLF) sectors as shown in Table 13.1 (that is, using the upper left 6 ×4 subblock of data).

13.2

On Planet Claire, markets are highly volatile. Here’s some recent historical data:

Year-Month

Hair Products

Cosmetics

Cash

2007-04

1.0

2.0

1.0

2007-03

2.0

2.0

1.0

2007-02

2.0

0.5

1.0

2007-01

0.5

2.0

1.0

Find every portfolio on Planet Claire’s efficient frontier.

13.3

What is the dual of (13.5)?

Notes

The portfolio selection problem originates with Markowitz (1959). He won the 1990 Nobel prize in Economics for this work. In its original formulation, risk is modeled by the variance of the portfolio’s value rather than the absolute deviation from the mean considered here. We will discuss the quadratic formulation later in Chapter 24.

The MAD risk measure we have considered in this chapter has many nice properties the most important of which is that it produces portfolios that are guaranteed not to be stochastically dominated (to second order) by other portfolios. Many risk measures fail to possess this important property. See Ruszczyński and Vanderbei (2003) for details.

Bibliography

Markowitz, H. (1959). Portfolio selection: Efficient diversification of investments. New York: Wiley.

Ruszczyński, A., and Vanderbei, R. (2003). http://​www.​princeton.​edu/​~rvdb/​tex/​lpport/​lpport8.​pdf Frontiers of stochastically nondominated portfolios. Econometrica, 71(4), 1287–1297.

Footnotes

1

In this chapter, we assume a modest familiarity with the ideas and notations of probability: the symbol $$\mathbb{E}$$ denotes expected value, which means that, if R is a random variable that takes values R(1), R(2), …, R(T) with equal probability, then

Part 2
Network-Type Problems

Allow me to say, …, that the arguments with which you have supported this extraordinary application have been as frivolous as the application was ill-judged. — J. Austen

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_14

14. Network Flow Problems

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.​1007/​978-1-4614-7630-6_​26

Many linear programming problems can be viewed as a problem of minimizing the “transportation” cost of moving materials through a network to meet demands for material at various locations given sources of material at other locations. Such problems are called network flow problems. They form the most important special class of linear programming problems. Transportation, electric, and communication networks provide obvious examples of application areas. Less obvious, but just as important, are applications in facilities location, resource management, financial planning, and others.

In this chapter we shall formulate a special type of linear programming problem called the minimum-cost network flow problem. It turns out that the simplex method when applied to this problem has a very simple description and some important special properties. Implementations that exploit these properties benefit dramatically.

1 Networks

A network consists of two types of objects: nodes and arcs. We shall let $$\mathcal{N}$$ denote the set of nodes . We let m denote the number of nodes (i.e., the cardinality of the set $$\mathcal{N}$$).

The nodes are connected by arcs . Arcs are assumed to be directed. This means that an arc connecting node i to node j is not the same as an arc connecting node j to node i. For this reason, we denote arcs using the standard mathematical notation for ordered pairs. That is, the arc connecting node i to node j is denoted simply as (i, j). We let $$\mathcal{A}$$ denote the set of all arcs in the network. This set is a subset of the set of all possible arcs:

In typical networks, the set $$\mathcal{A}$$ is much smaller than the set of all arcs. In fact, usually each node is only connected to a handful of “nearby” nodes.

The pair $$(\mathcal{N},\mathcal{A})$$ is called a network . It is also sometimes called a graph or a digraph (to emphasize the fact that the arcs are directed). Figure 14.1 shows a network having 7 nodes and 14 arcs.

Fig. 14.1

A network having 7 nodes and 14 arcs. The numbers written next to the nodes denote the supply at the node (negative values indicate demands; missing values indicate no supply or demand).

To specify a network flow problem, we need to indicate the supply of (or demand for) material at each node. So, for each $$i \in \mathcal{N}$$, let b i denote the amount of material being supplied to the network at node i. We shall use the convention that negative supplies are in fact demands. Hence, our problem will be to move the material that sits at the supply nodes over to the demand nodes. The movements must be along the arcs of the network (and adhering to the directions of the arcs). Since, except for the supply and demand, there is no other way for material to enter or leave the system, it follows that the total supply must equal the total demand for the problem to have a feasible solution. Hence, we shall always assume that

To help us decide the paths along which materials should move, we assume that each arc, say, (i, j), has associated with it a cost c ij that represents the cost of shipping one unit from i to j directly along arc (i, j). The decision variables then are how much material to ship along each arc. That is, for each $$(i,j) \in \mathcal{A}$$, x ij will denote the quantity shipped directly from i to j along arc (i, j). The objective is to minimize the total cost of moving the supply to meet the demand:

As we mentioned before, the constraints on the decision variables are that they ensure flow balance at each node. Let us consider a fixed node, say, $$k \in \mathcal{N}$$. The total flow into node k is given by

Similarly, the total flow out from node k is

The difference between these two quantities is the net inflow, which must be equal to the demand at the node. Hence, the flow balance constraints can be written as

Finally, the flow on each arc must be nonnegative (otherwise it would be going in the wrong direction):

Figure 14.2 shows cost information for the network shown in Figure 14.1. In matrix notation, the problem can be written as follows:

(14.1)

where

Fig. 14.2

The costs on the arcs for the network in Figure 14.1.

In network flow problems, the constraint matrix A is called the node–arc incidence matrix .

The network flow problem differs from our usual standard form linear programming problem in two respects: (1) it is a minimization instead of a maximization and (2) the constraints are equalities instead of inequalities. Nonetheless, we have studied before how duality applies to problems in nonstandard form. The dual of (14.1) is

Written in network notation, the dual is

Finally, it is not hard to check that the complementarity conditions (to be satisfied by an optimal primal–dual solution pair) are

We shall often refer to the primal variables as primal flows .

2 Spanning Trees and Bases

Network flow problems can be solved efficiently because the basis matrices have a special structure that can be described nicely in terms of the network. In order to explain this structure, we need to introduce a number of definitions.

First of all, an ordered list of nodes (n ₁, n ₂, …, n k ) is called a path in the network if each adjacent pair of nodes in the list is connected by an arc in the network. It is important to note that we do not assume that the arcs point in any particular direction. For example, for nodes n i and n i+1, there must be an arc in the network. It could run either from n i to n i+1 or from n i+1 to n i . (One should think about one-way roads—even though cars can only go one way, pedestrians are allowed to walk along the path of the road in either direction.) A network is called connected if there is a path connecting every pair of nodes (see Figure 14.3). For the remainder of this chapter, we make the following assumption:

Fig. 14.3

The network on the left is connected whereas the one on the right is not.

Assumption. The network is connected.

For any arc (i, j), we refer to i as its tail and j as its head .

A cycle is a path in which the last node coincides with the first node. A network is called acyclic if it does not contain any cycles (see Figure 14.4).

Fig. 14.4

The network on the left contains a cycle whereas the one on the right is acyclic.

A network is a tree if it is connected and acyclic (see Figure 14.5). A network $$(\tilde{\mathcal{N}},\tilde{\mathcal{A}})$$ is called a subnetwork of $$(\mathcal{N},\mathcal{A})$$ if $$\tilde{\mathcal{N}}\subset \mathcal{N}$$ and $$\tilde{\mathcal{A}}\subset \mathcal{A}$$. A subnetwork $$(\tilde{\mathcal{N}},\tilde{\mathcal{A}})$$ is a spanning tree if it is a tree and $$\tilde{\mathcal{N}} = \mathcal{N}$$. Since a spanning tree’s node set coincides with the node set of the underlying network, it suffices to refer to a spanning tree by simply giving its arc set.

Fig. 14.5

The network on the left is a tree whereas the two on the right not—they fail in the first case by being disconnected and in the second by containing a cycle.

Given a network flow problem, any selection of primal flow values that satisfies the balance equations at every node will be called a balanced flow . It is important to note that we do not require the flows to be nonnegative to be a balanced flow. That is, we allow flows to go in the wrong direction. If all the flows are nonnegative, then a balanced flow is called a feasible flow . Given a spanning tree, a balanced flow that assigns zero flow to every arc not on the spanning tree will be called a tree solution . Consider, for example, the tree shown in Figure 14.6. The numbers shown on the arcs of the spanning tree give the tree solution corresponding to the supplies/demands shown in Figure 14.1. They were obtained by starting at the “leaves” of the tree and working “inward.” For instance, the flows could be solved for successively as follows:

It is easy to see that this process always works. The reason is that every tree must have at least one leaf node, and deleting a leaf node together with the edge leading into it produces a subtree.

Fig. 14.6

The fat arcs show a spanning tree for the network in Figure 14.1. The numbers shown on the arcs of the spanning tree are the primal flows, the numbers shown next to the nodes are the dual variables, and the numbers shown on the arcs not belonging to the spanning tree are the dual slacks.

The above computation suggests that spanning trees are related to bases in the simplex method. Let us pursue this idea. Normally, a basis is an invertible square submatrix of the constraint matrix. But for incidence matrices, no such submatrix exists. To see why, note that if we sum together all the rows of A, we get a row vector of all zeros (since each column of A has exactly one + 1 and one − 1). Of course, every square submatrix of A has this same property and so is singular. In fact, we shall show in a moment that for a connected network, there is exactly one redundant equation (i.e., the rank of A is exactly m − 1).

Let us select some node, say, the last one, and delete the flow-balance constraint associated with this node from the constraints defining the problem (since it is redundant anyway). Let’s call this node the root node . Let $$\tilde{A}$$ denote the incidence matrix A without the row corresponding to the root node (i.e., the last row), and let $$\tilde{b}$$ denote the supply/demand vector with the last entry deleted. The most important property of network flow problems is summarized in the following theorem:

Theorem 14.1.

A square submatrix of $$\tilde{A}$$ is a basis if and only if the arcs to which its columns correspond form a spanning tree.

Rather than presenting a formal proof of this theorem, it is more instructive to explain the idea using the example we’ve been studying. Therefore, consider the spanning tree shown in Figure 14.6, and let B denote the square submatrix of $$\tilde{A}$$ corresponding to this tree. The matrix B is invertible if and only if every system of equations of the form

has a unique solution. This is exactly the type of equation that we already solved to find the tree solution associated with the spanning tree:

We solved this system of equations by looking at the spanning tree and realizing that we could work our way to a solution by starting with the leaves and working inward. This process amounts to a permutation of the rows and columns of B to get a lower triangular matrix. Indeed, for the calculations given above, we have permuted the rows by P and the columns by Q to get

The fact that B is invertible is now immediately apparent from the fact that the permuted matrix is lower triangular. In fact, it has only + 1’s and − 1’s on the diagonal. Therefore, we can solve systems of equations involving B without ever having to do any divisions. Also, since the off-diagonal entries are also ± 1’s, it follows that we don’t need to do any multiplications either. Every system of equations involving the matrix B can be solved by a simple sequence of additions and subtractions.

We have shown that, given a spanning tree, the submatrix of $$\tilde{A}$$ consisting of the columns corresponding to the arcs in the spanning tree is a basis. The converse direction (which is less important to our analysis) is relegated to an exercise (see Exercise 14.12).

Not only is there a primal solution associated with any basis but also there is a dual solution. Hence, corresponding to any spanning tree there is a dual solution. The dual solution consists of two types of variables: the y i ’s and the z ij ’s. These variables must satisfy the dual feasibility conditions:

By complementarity, z ij  = 0 for each (i, j) in the spanning tree $$\mathcal{T}$$. Hence,

Since a spanning tree on m nodes has m − 1 arcs (why?), these equations define a system of m − 1 equations in m unknowns. But don’t forget that there was a redundant equation in the primal problem, which we associated with a specific node called the root node. Removing that equation and then looking at the dual, we see that there is not really a dual variable associated with the root node. Or equivalently, we can just say that the dual variable for the root node is zero. Making this assignment, we get m equations in m unknowns. These equations can be solved by starting at the root node and working down the tree.

For example, let node “g” be the root node in the spanning tree in Figure 14.6. Starting with it, we compute the dual variables as follows:

Now that we know the dual variables, the dual slacks for the arcs not in the spanning tree $$\mathcal{T}$$ can be computed using

(which is just the dual feasibility condition solved for z ij ). These values are shown on the nontree arcs in Figure 14.6.

From duality theory, we know that the current tree solution is optimal if all the flows are nonnegative and if all the dual slacks are nonnegative. The tree solution shown in Figure 14.6 satisfies the first condition but not the second. That is, it is primal feasible but not dual feasible. Hence, we can apply the primal simplex method to move from this solution to an optimal one. We take up this task in the next section.

3 The Primal Network Simplex Method

Each of the variants of the simplex method presented in earlier chapters of this book can be applied to network flow problems. It would be overkill to describe them all here in the context of networks. However, they are all built on two simple algorithms: the primal simplex method (for problems that are primal feasible) and the dual simplex method (for problems that are dual feasible). We discuss them both in detail.

We shall describe the primal network simplex method by continuing with our example. As mentioned above, the tree shown in Figure 14.6 is primal feasible but not dual feasible. The basic idea that defines the primal simplex method is to pick a nontree arc that is dual infeasible and let it enter the tree (i.e., become basic) and then readjust everything so that we still have a tree solution.

The First Iteration. For our first pivot, we let arc (a,c) enter the tree using a primal pivot . In a primal pivot, we add flow to the entering variable, keeping all other nontree flows set to zero and adjusting the tree flows appropriately to maintain flow balance. Given any spanning tree, adding an extra arc must create a cycle (why?). Hence, the current spanning tree together with the entering arc must contain a cycle. The flows on the cycle must change to accommodate the increasing flow on the entering arc. The flows on the other tree arcs remain unchanged. In our example, the cycle is: “a”, “c”, “b”, “f”. This cycle is shown in Figure 14.7 with flows adjusted to take into account a flow of t on the entering arc. As t increases, eventually the flow on arc (f,b) decreases to zero. Hence, arc (f,b) is the leaving arc. Updating the flows is easy; just take t = 3 and adjust the flows appropriately.

Fig. 14.7

The cycle produced by including the entering arc with the spanning tree. As the flow t on the entering arc increases, eventually the flow on arc (f,b) becomes zero (when t = 3). Hence, arc (f,b) is the leaving arc.

With a little thought, one realizes that the selection rule for the leaving arc in a primal pivot is as follows:

Leaving arc selection rule:

The leaving arc must be oriented along the cycle in the reverse direction from the entering arc, and

Among all such arcs, it must have the smallest flow.

Also, the flows on the cycle get updated as follows:

Primal flows update:

Flows oriented in the same direction as the leaving arc are decreased by the amount of flow that was on the leaving arc whereas flows in the opposite direction are increased by this amount.

The next issue is how to update the dual variables. To this end, note that if we delete the leaving arc from the spanning tree (without concurrently adding the entering arc), we disconnect it into two disjoint trees. In our example, one tree contains nodes “a”, “d” and “f” while the second tree contains the other nodes. Figure 14.8 shows the two disjoint trees. Recalling that the dual variables are calculated starting with the root node and working up the spanning tree, it is clear that the dual variables on the subtree containing the root node remain unchanged, whereas those on the other subtree must change. For the current pivot, the other subtree consists of nodes “a”, “d”, and “f”. They all get incremented by the same fixed amount, since the only change is that the arc by which we bridged from the root-containing tree to this other tree has changed from the leaving arc to the entering arc. Looking at node “a” and using tildes to denote values after being changed, we see that

whereas

Combining these two equations, we get

That is, the dual variable at node “a” gets decremented by z ac = −9. Of course, all of the dual variables on this subtree get decremented by this same amount. In general, the dual variable update rule can be stated as follows:

Dual variables update:

If the entering arc crosses from the root-containing tree to the non-root-containing tree, then increase all dual variables on the non-root-containing tree by the dual slack of the entering arc.

Otherwise, decrease these dual variables by this amount.

Fig. 14.8

The two disjoint trees. Primal and dual values that remained unchanged are shown, whereas those that need to be updated are shown as question marks.

Finally, we must update the dual slacks. The only dual slacks that change are those that span across the two trees since, for these nodes, either the head or the tail dual variable changes, while the other does not. Those that span the two subtrees in the same direction as the entering arc must be decreased by z ac, whereas those that bridge the two trees in the opposite direction must be increased by this amount. For our example, six nontree arcs, (f,g), (f,b), (f,c), (d,b), (d,e), and (a,e), span in the same direction as the entering arc. They all must be decreased by − 9. That is, they must be increased by 9. For example, the dual slack on arc (f,c) changes from − 5 to 4. Only one arc, (b,a), spans in the other direction. It must be decreased by 9. The updated solution is shown in Figure 14.9. The general rule for updating the dual slacks is as follows:

Dual slacks update:

The dual slacks corresponding to those arcs that bridge in the same direction as the entering arc get decremented by the old dual slack on the entering arc, whereas those that correspond to arcs bridging in the opposite direction get incremented by this amount.

Fig. 14.9

The tree solution at the end of the first iteration.

The Second Iteration. The tree solution shown in Figure 14.9 has only one remaining infeasibility: z ba = −10. Arc (b,a) must therefore enter the spanning tree. Adding it, we create a cycle consisting of nodes “a”, “b”, and “c”. The leaving arc must be pointing in the opposite direction from the entering arc. Here, there is only one such arc, (b,c). It must be the leaving arc. The leaving arc’s flow decreases from 3 to 0. The flow on the other two cycle arcs must increase by 3 to preserve flow balance.

The two subtrees formed by removing the leaving arc are shown in Figure 14.10 The dual variables on the non-root-containing subtree get incremented by the dual slack on the entering arc z ba = −10. The dual slacks for the spanning arcs also change by 10 either up or down depending on which way they bridge the two subtrees. The resulting tree solution is shown in Figure 14.11.

Fig. 14.10

The two disjoint subtrees arising in the second iteration.

Fig. 14.11

The tree solution at the end of the second iteration. To get from the spanning tree in Figure 14.9 to here, we let arc (b,a) enter and arc (b,c) leave.

The Third and Final Iteration. The tree solution shown in Figure 14.11 has one infeasibility: z fb = −1. Hence, arc (f,b) must enter the spanning tree. The leaving arc must be (f,a). Leaving the details of updating to the reader, the resulting tree solution is shown in Figure 14.12. It is both primal and dual feasible—hence optimal.

Fig. 14.12

The tree solution at the end of the third iteration. To get from the spanning tree in Figure 14.11 to here, we let arc (f,b) enter and arc (f,a) leave. This tree solution is the optimal solution to the problem.

4 The Dual Network Simplex Method

In the previous section, we developed simple rules for the primal network simplex method, which is used in situations where the tree solution is primal feasible but not dual feasible. When a tree solution is dual feasible but not primal feasible, then the dual network simplex method can be used. We shall define this method now. Consider the tree solution shown in Figure 14.13. It is dual feasible but not primal feasible (since x db < 0). The basic idea that defines the dual simplex method is to pick a tree arc that is primal infeasible and let it leave the spanning tree (i.e., become nonbasic) and then readjust everything to preserve dual feasibility.

Fig. 14.13

A tree solution that is dual feasible but not primal feasible.

The First Iteration. For the first iteration, we need to let arc (d,b) leave the spanning tree using a dual pivot , which is defined as follows. Removing arc (d,b) disconnects the spanning tree into two disjoint subtrees. The entering arc must be one of the arcs that spans across the two subtrees so that it can reconnect them into a spanning tree. That is, it must be one of

See Figure 14.14. To see how to decide which it must be, we need to consider carefully the impact of each possible choice.

Fig. 14.14

The two subtrees for the first pivot of the dual simplex method.

To this end, let us consider the general situation. As mentioned above, the spanning tree with the leaving arc removed consists of two disjoint trees. The entering arc must reconnect these two trees.

First, consider a reconnecting arc that connects in the same direction as the leaving arc. When we add flow to this prospective entering arc, we will have to decrease flow on the leaving arc to maintain flow balance. Therefore, the leaving arc’s flow, which is currently negative, can’t be raised to zero. That is, the leaving arc can’t leave. This is no good.

Now suppose that the reconnecting arc connects in the opposite direction. If it were to be the entering arc, then its dual slack would drop to zero. All other reconnecting arcs pointing in the same direction would drop by the same amount. To maintain nonnegativity of all the others, we must pick the one that drops the least. We can summarize the rule as follows:

Entering arc selection rule:

The entering arc must bridge the two subtrees in the opposite direction from the leaving arc, and

Among all such arcs, it must have the smallest dual slack.

In our example, all bridging arcs point in the opposite direction from the leaving arc. The one with the smallest dual slack is (g,e) whose slack is z ge = 9. This arc must be the entering arc.

We have now determined both the entering and leaving arcs. Hence, the new spanning tree is determined and therefore, in principle, all the variables associated with this new spanning tree can be computed. Furthermore, the rules for determining the new values by updating from the previous ones are the same as in the primal network simplex method. The resulting tree solution is shown in Figure 14.15.

Fig. 14.15

The tree solution after the first pivot.

The Second Iteration. For the second pivot, there are two choices for the leaving arc: (g,b) and (d,e). Using the most infeasible, we choose (d,e). We remove this arc from the spanning tree to produce two subtrees. One of the subtrees consists of just the node “d” all by itself while the other subtree consists of the rest of the nodes. Remembering that the reconnecting arc must bridge the two subtrees in the opposite direction, the only choice is (a,d). So this arc is the entering arc. Making the pivot, we arrive at the optimal tree solution shown in Figure 14.12.

5 Putting It All Together

As we saw in Chap. 5, for linear programming the primal and the dual simplex methods form the foundation on which one can build a few different variants of the simplex method. The same is true here in the context of network flows.

For example, one can build a two-phased procedure in which one first uses the dual network simplex method (with costs artificially and temporarily altered to ensure dual feasibility of an initial tree solution) to find a primal feasible solution and then uses the primal network simplex method to move from the feasible solution to an optimal one.

Alternatively, one can use the primal network simplex method (with supplies temporarily altered to ensure primal feasibility of an initial tree solution) to find a dual feasible solution and then use the dual network simplex method (with the original supplies) to move from the dual feasible solution to an optimal one.

Finally, as described for linear programming in Chap. 7, one can define a parametric self-dual method in which primal pivots and dual pivots are intermingled as needed so as to reduce a perturbation parameter μ from ∞ to zero.

Since there is nothing new in how one builds the network versions of these algorithms from the basic primal and dual simplex pivots, we don’t go through any examples here. Instead, we just mention one final observation about the dual variables, the y i ’s. Namely, they are not needed anywhere in the performance of a primal or a dual pivot. Hence, their calculation is entirely optional and can be skipped altogether or simply deferred to the end.

For completeness, we end this section by giving a step-by-step description of the self-dual network simplex method. The steps are as follows:

1.

Identify a spanning tree—any one will do (see Exercise 14.14). Also identify a root node.

2.

Compute initial primal flows on the tree arcs by assuming that nontree arcs have zero flow and the total flow at each node must be balanced. For this calculation, the computed primal flows may be negative. In this case, the initial primal solution is not feasible. The calculation is performed working from leaf nodes inward.

3.

Compute initial dual values by working out from the root node along tree arcs using the formula

which is valid on tree arcs, since the dual slacks vanish on these arcs.

4.

Compute initial dual slacks on each nontree arc using the formula

Again, some of the z ij ’s might be nonnegative. This is okay (for now), but it is important that they satisfy the above equality.

5.

Perturb each primal flow and each dual slack that has a negative initial value by adding a positive scalar μ to each such value.

6.

Identify a range μ min  ≤ μ ≤ μ max over which the current solution is optimal (on the first iteration, μ max will be infinite).

7.

Check the stopping rule: if μ min  ≤ 0, then set μ = 0 to recover an optimal solution. While not optimal, perform each of the remaining steps and then return to recheck this condition.

8.

Select an arc associated with the inequality μ min  ≤ μ (if there are several, pick one arbitrarily). If this arc is a nontree arc, then the current pivot is a primal pivot. If, on the other hand, it is a tree arc, then the pivot is a dual pivot.

a.

If the pivot is a primal pivot, the arc identified above is the entering arc. Identify the associated leaving arc as follows. First, add the entering arc to the tree. With this arc added, there must be a cycle consisting of the entering arc and other tree arcs. The leaving arc is chosen from those arcs on the cycle that go in the opposite direction from the entering arc and having the smallest flow among all such arcs (evaluated at μ = μ min ).

b.

If the pivot is a dual pivot, the arc identified above is the leaving arc. Identify the associated entering arc as follows. First, delete the leaving arc from the tree. This deletion splits the tree into two subtrees. The entering arc must bridge these two trees in the opposite direction to the leaving arc, and, among such arcs, it must be the one with the smallest dual slack (evaluated at μ = μ min ).

9.

Update primal flows as follows. Add the entering arc to the tree. This addition creates a cycle containing both the entering and leaving arcs. Adjust the flow on the leaving arc to zero, and then adjust the flows on each of the other cycle arcs as necessary to maintain flow balance.

10.

Update dual variables as follows. Delete the leaving arc from the old tree. This deletion splits the old tree into two subtrees. Let $$\mathcal{T}_{u}$$ denote the subtree containing the tail of the entering arc, and let $$\mathcal{T}_{v}$$ denote the subtree containing its head. The dual variables for nodes in $$\mathcal{T}_{u}$$ remain unchanged, but the dual variables for nodes in $$\mathcal{T}_{v}$$ get incremented by the old dual slack on the entering arc.

11.

Update dual slacks as follows. All dual slacks remain unchanged except for those associated with nontree arcs that bridge the two subtrees $$\mathcal{T}_{u}$$ and $$\mathcal{T}_{v}$$. The dual slacks corresponding to those arcs that bridge in the same direction as the entering arc get decremented by the old dual slack on the entering arc, whereas those that correspond to arcs bridging in the opposite direction get incremented by this amount.

As was said before and should now be clear, there is no need to update the dual variables from one iteration to the next; that is, step 10 can be skipped.

6 The Integrality Theorem

In this section, we consider network flow problems for which all the supplies and demands are integers. Such problems are called network flow problems with integer data. As we explained in Sect. 14.2, for network flow problems, basic primal solutions are computed without any multiplication or division. The following important theorem follows immediately from this property:

Theorem 14.2.

Integrality Theorem. For network flow problems with integer data, every basic feasible solution and, in particular, every basic optimal solution assigns integer flow to every arc.

This theorem is important because many real-world network flow problems have integral supplies/demands and require their solutions to be integral too. This integrality restriction typically occurs when one is shipping indivisible units through a network. For example, it wouldn’t make sense to ship one third of a car from an automobile assembly plant to one dealership with the other two thirds going to another dealership.

Problems that are linear programming problems with the additional stipulation that the optimal solution values must be integers are called integer programming problems . Generally speaking, these problems are much harder to solve than linear programming problems (see Chap. 23). However, if the problem is a network flow problem with integer data, it can be solved efficiently using the simplex method to compute a basic optimal solution, which the integrality theorem tells us will be integer valued.

6.1 König’s Theorem

In addition to its importance in real-world optimization problems, the integrality theorem also has many applications to the branch of mathematics called combinatorics. We illustrate with just one example.

Theorem 14.3.

König’s Theorem. Suppose that there are n girls and n boys, that every girl knows exactly k boys, and that every boy knows exactly k girls. Then n marriages can be arranged with everybody knowing his or her spouse.

Before proving this theorem it is important to clarify its statement by saying that the property of “knowing” is symmetric (for example, knowing in the biblical sense). That is, if a certain girl knows a certain boy, then this boy also knows this girl.

Proof.

Consider a network with nodes g ₁, g ₂, …, g n , b ₁, b ₂, …, b n and an arc from g i to b j if girl i and boy j know each other. Assign one unit of supply to each girl node and a unit of demand to each boy node. Assign arbitrary objective coefficients to create a well-defined network flow problem. The problem is guaranteed to be feasible: just put a flow of 1∕k on each arc (the polygamists in the group might prefer this nonintegral solution). By the integrality theorem, the problem has an integer-valued solution. Clearly, the flow on each arc must be either zero or one. Also, each girl node is the tail of exactly one arc having a flow of one. This arc points to her intended mate.

Exercises

In solving the following problems, the network pivot tool can be used to check your arithmetic:

www.​princeton.​edu/​rvdb/​JAVA/​network/​nettool/​netsimp.​html

14.1

Consider the following network flow problem:

Numbers shown above the nodes are supplies (negative values represent demands) and numbers shown above the arcs are unit shipping costs. The darkened arcs form a spanning tree.

(a)

Compute primal flows for each tree arc.

(b)

Compute dual variables for each node.

(c)

Compute dual slacks for each nontree arc.

14.2

Consider the tree solution for the following minimum cost network flow problem:

The numbers on the tree arcs represent primal flows while numbers on the nontree arcs are dual slacks.

(a)

Using the largest–coefficient rule in the dual network simplex method, what is the leaving arc?

(b)

What is the entering arc?

(c)

After one pivot, what is the new tree solution?

14.3

Consider the following network flow problem:

The numbers above the nodes are supplies (negative values represent demands) and numbers shown above the arcs are unit shipping costs. The darkened arcs form a spanning tree.

(a)

Compute primal flows for each tree arc.

(b)

Compute dual variables for each node.

(c)

Compute dual slacks for each nontree arc.

14.4

Consider the tree solution for the following minimum cost network flow problem:

The numbers on the tree arcs represent primal flows while numbers on the nontree arcs are dual slacks.

(a)

Using the largest–coefficient rule in the primal network simplex method, what is the entering arc?

(b)

What is the leaving arc?

(c)

After one pivot, what is the new tree solution?

14.5

Consider the tree solution for the following minimum cost network flow problem:

The numbers on the tree arcs represent primal flows while numbers on the nontree arcs are dual slacks.

(a)

Using the largest–coefficient rule in the dual network simplex method, what is the leaving arc?

(b)

What is the entering arc?

(c)

After one pivot, what is the new tree solution?

14.6

Solve the following network flow problem starting with the spanning tree shown.

The numbers displayed next to nodes are supplies(+)/demands(−). Numbers on arcs are costs. Missing data should be assumed to be zero. The bold arcs represent an initial spanning tree.

14.7

Solve Exercise 2.11 using the self-dual network simplex method.

14.8

Using today’s date (MMYY) for the seed value, solve ten problems using the network simplex pivot tool:

www.​princeton.​edu/​rvdb/​JAVA/​network/​challenge/​netsimp.​html

14.9

Consider the following tree solution for a minimum cost network flow problem:

As usual, bold arcs represent arcs on the spanning tree, numbers next to the bold arcs are primal flows, numbers next to non-bold arcs are dual slacks, and numbers next to nodes are dual variables.

(a)

For what values of μ is this tree solution optimal?

(b)

What are the entering and leaving arcs?

(c)

After one pivot, what is the new tree solution?

(d)

For what values of μ is the new tree solution optimal?

14.10

Consider the following tree solution for a minimum cost network flow problem:

(a)

For what values of μ is this tree solution optimal?

(b)

What are the entering and leaving arcs?

(c)

After one pivot, what is the new tree solution?

(d)

For what values of μ is the new tree solution optimal?

14.11

Consider the following minimum cost network flow problem

As usual, the numbers on the arcs represent the flow costs and numbers at the nodes represent supplies (demands are shown as negative supplies). The arcs shown in bold represent a spanning tree. If the solution corresponding to this spanning tree is optimal prove it, otherwise find an optimal solution using this tree as the initial spanning tree.

14.12

Suppose that a square submatrix of $$\tilde{A}$$ is invertible. Show that the arcs corresponding to the columns of this submatrix form a spanning tree.

14.13

Show that a spanning tree on m nodes must have exactly m − 1 arcs.

14.14

Define an algorithm that takes as input a network and either finds a spanning tree or proves that the network is not connected.

14.15

Give an example of a minimum-cost network flow problem with all arc costs positive and the following counterintuitive property: if the supply at a particular source node and the demand at a particular sink node are simultaneously reduced by one unit, then the optimal cost increases.

14.16

Consider a possibly disconnected network $$(\mathcal{N},\mathcal{A})$$. Two nodes i and j in $$\mathcal{N}$$ are said to be connected if there is a path from i to j (recall that paths can traverse arcs backwards or forwards). We write i ∼ j if i and j are connected.

(a)

Show that “ ∼ ” defines an equivalence relation . That is, it has the following three properties:

(i)

(Reflexivity) for all $$i \in \mathcal{N}$$, i ∼ i;

(ii)

(Symmetry) for all $$i,j \in \mathcal{N}$$, i ∼ j implies that j ∼ i;

(iii)

(Transitivity) for all $$i,j,k \in \mathcal{N}$$, i ∼ j and j ∼ k implies that i ∼ k.

Using the equivalence relation, we can partition $$\mathcal{N}$$ into a collection of subsets of equivalence classes $$\mathcal{N}_{1},\mathcal{N}_{2},\ldots,\mathcal{N}_{k}$$ such that two nodes are connected if and only if they belong to the same subset. The number k is called the number of connected components .

(a)

Show that the rank of the node–arc incidence matrix A is exactly m − k (recall that m is the number of rows of A).

14.17

One may assume without loss of generality that every node in a minimum cost network flow problem has at least two arcs associated with it. Why?

14.18

The sum of the dual slacks around any cycle is a constant. What is that constant?

14.19

Planar Networks. A network is called planar if the nodes and arcs can be laid out on the two-dimensional plane in such a manner that no two arcs cross each other (it is allowed to draw the arcs as curves if necessary). All of the networks encountered so far in this chapter have been planar. Associated with each planar network is a geometrically defined dual network. The purpose of this problem is to establish the following interesting fact:

A dual network simplex pivot is precisely a primal network simplex method applied to the dual network.

Viewed geometrically, the nodes of a planar graph are called vertices and the arcs are called edges. Consider a specific connected planar network. If one were to delete the vertices and the edges from the plane, one would be left with a disjoint collection of subsets of the plane. These subsets are called faces. Note that there is one unbounded face. It is a face just like the other bounded ones. An example of a connected planar network with its faces labeled A through D is shown in Figure 14.16.

Fig. 14.16

The primal network has nodes “a” through “f”. The corresponding dual network has nodes “A” through “D” (node “A” is “at infinity”). A primal spanning tree is shown. It consists of five arcs: (a,b), (f,b), (b,e), (e,d), and (c,d). The corresponding dual spanning tree consists of three arcs: (B,A), (A,C), and (D,A). Primal costs are shown along the primal arcs and supplies/demands are shown at the primal nodes.

Dual nodes. Associated with each connected planar network is a dual network defined by interchanging vertices and faces. That is, place a dual vertex in the center of each primal face. Note: the dual vertex corresponding to the unbounded primal face could be placed anywhere in the unbounded face but we choose to put it at infinity. In this way, dual edges (defined next) that have a head or a tail at this node can run off to infinity in any direction.

Dual arcs. Connect with a dual edge any pair of dual nodes whose corresponding primal faces share an edge. Each dual edge crosses exactly one primal edge. The directionality of the dual edge is determined as follows: first, place a vector along the corresponding primal edge pointing in the direction of the primal arc, and then rotate it counterclockwise until it is tangent to the dual edge. The vector now defines the direction for the dual arc.

Dual spanning tree. Consider a spanning tree on the primal network and suppose that a primal–dual tree solution is given. We define a spanning tree on the dual network as follows. A dual edge is on the dual network’s spanning tree if and only if the corresponding primal edge is not on the primal network’s spanning tree.

Dual flows and dual dual-slacks. The numerical arc data for the dual network is inherited directly from the primal. That is, flows on the dual tree arcs are exactly equal to the dual slacks on the associated primal nontree arcs. And, the dual slacks on the dual nontree arcs are exactly equal to the primal flows on the associated primal tree arcs. Having specified numerical data on the arcs of the dual network, it is fairly straightforward to determine values for supplies/demands at the nodes and shipping costs along the arcs that are consistent with these numerical values.

(a)

Which of the following networks are planar:

(b)

A network is called complete if there is an arc between every pair of nodes. If a complete network with m nodes is planar, then every network with m nodes is planar. Prove it.

(c)

Show that a nonplanar network must have five or more nodes.

(d)

As always, let m denote the number of nodes and let n denote the number of arcs in a network. Let f denote the number of faces in a planar network. Show by induction on f that m = n − f + 2.

(e)

Show that the dual spanning tree defined above is in fact a spanning tree.

(f)

Show that a dual pivot for a minimum cost network flow problem defined on the primal network is precisely the same as a primal pivot for the corresponding network flow problem on the dual network.

(g)

Using the cost and supply/demand information given for the primal problem in Figure 14.16, write down the primal problem as a linear programming problem.

(h)

Write down the dual linear programming problem that one derives algebraically from the primal linear programming problem.

(i)

Using the spanning tree shown in Figure 14.16, compute the primal flows, dual variables, and dual slacks for the network flow problem associated with the primal network.

(j)

Write down the flow and slacks for the network flow problem associated with the dual network.

(k)

Find arc costs and node supplies/demands for the dual network that are consistent with the flows and slacks just computed.

(l)

Write down the linear programming problem associated with the network flow problem on the dual network.

Notes

The classical reference is Ford and Fulkerson (1962). More recent works include the books by Christofides (1975), Lawler (1976), Bazaraa et al. (1977), Kennington and Helgason (1980), Jensen and Barnes (1980), Bertsekas (1991), and Ahuja et al. (1993).

The two “original” algorithms for solving minimum-cost network flow problems are the network simplex method developed by Dantzig (1951a) and the primal–dual method developed by Ford and Fulkerson (1958). The self-dual algorithm described in this chapter is neither of these. In fact, it resembles the “out-of-kilter” method described by Ford and Fulkerson (1962).

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Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_15

15. Applications

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In this chapter, we discuss briefly the most important applications of network flow problems.

1 The Transportation Problem

The network flow problem, when thought of as representing the shipment of goods along a transportation network, is called the transshipment problem. An important special case is when the set of nodes $$\mathcal{N}$$ can be partitioned into two sets $$\mathcal{S}$$ and $$\mathcal{D}$$,

such that every arc in $$\mathcal{A}$$ has its tail in $$\mathcal{S}$$ and its head in $$\mathcal{D}$$. The nodes in $$\mathcal{S}$$ are called source (or supply) nodes, while those in $$\mathcal{D}$$ are called destination (or demand) nodes. Such graphs are called bipartite graphs (see Figure 15.1). A network flow problem on such a bipartite graph is called a transportation problem.

FIGURE 15.1

A bipartite graph—the network for a transportation problem.

In order for a transportation problem to be feasible, the supply must be nonnegative at every supply node, and the demand must be nonnegative at every demand node. That is,

When put on paper, a bipartite graph has the annoying property that the arcs tend to cross each other many times. This makes such a representation inconvenient for carrying out the steps of the network simplex method. But there is a nice, uncluttered, tabular representation of a bipartite graph that one can use when applying the simplex method. To discover this tabular representation, first suppose that the graph is laid out as shown in Figure 15.2. Now if we place the supplies and demands on the nodes and the costs at the kinks in the arcs, then we get, for example, the following simple tabular representation of a transportation problem:

(15.1)

(the asterisks represent nonexistent arcs). The iterations of the simplex method can be written in this tabular format by simply placing the dual variables where the supplies and demands are and by placing the primal flows and dual slacks where the arc costs are. Of course, some notation needs to be introduced to indicate which cells are part of the current spanning tree. For example, the tree could be indicated by putting a box around the primal flow values. Here is a (nonoptimal) tree solution for the data given above:

(15.2)

(the solution to this problem is left as an exercise). In the case where every supply node is connected to every demand node, the problem is called the Hitchcock Transportation Problem. In this case, the equations defining the problem are especially simple. Indeed, if we denote the supplies at the supply nodes by r i , $$i \in \mathcal{S}$$, and if we denote the demands at the demand nodes by s j , $$j \in \mathcal{D}$$, then we can write the problem as

FIGURE 15.2

The bipartite graph from Figure 15.1 laid out in a rectangular fashion, with supplies and demands given at the nodes, and with costs given on the arcs.

2 The Assignment Problem

Given a set $$\mathcal{S}$$ of m people, a set $$\mathcal{D}$$ of m tasks, and for each $$i \in \mathcal{S}$$, $$j \in \mathcal{D}$$ a cost c ij associated with assigning person i to task j, the assignment problem is to assign each person to one and only one task in such a manner that each task gets covered by someone and the total cost of the assignments is minimized. If we let

then the objective function can be written as

The constraint that each person is assigned exactly one task can be expressed simply as

Also, the constraint that every task gets covered by someone is just

Except for the assumed integrality of the decision variables, x ij , the assignment problem is just a Hitchcock transportation problem in which the supply at every supply node (person) is one and the demand at every demand node (task) is also one. This Hitchcock transportation problem therefore is called the LP-relaxation of the assignment problem. It is easy to see that every feasible solution to the assignment problem is a feasible solution for its LP-relaxation. Furthermore, every integral feasible solution to the LP-relaxation is a feasible solution to the assignment problem. Since the network simplex method applied to the LP-relaxation produces an integral solution, it therefore follows that the method solves not only the LP-relaxation but also the assignment problem itself. We should note that this is a very special and important feature of the network simplex method. For example, had we used the primal–dual interior-point method to solve the LP-relaxation, there would be no guarantee that the solution obtained would be integral (unless the problem has a unique optimal solution, in which case any LP solver would find the same, integral answer—but typical assignment problems have alternate optimal solutions, and an interior-point method will report a convex combination of all of them).

3 The Shortest-Path Problem

Roughly speaking, the shortest-path problem is to find, well, the shortest path from one specific node to another in a network $$(\mathcal{N},\mathcal{A})$$. In contrast to earlier usage, the arcs connecting successive nodes on a path must point in the direction of travel. Such paths are sometimes referred to as directed paths. To determine a shortest path, we assume that we are given the length of each arc. To be consistent with earlier notations, let us assume that the length of arc (i, j) is denoted by c ij . Naturally, we assume that these lengths are nonnegative.

To find the shortest path from one node (say, s) to another (say, r), we will see that it is necessary to compute the shortest path from many, perhaps all, other nodes to r. Hence, we define the shortest-path problem as the problem of finding the shortest path from every node in $$\mathcal{N}$$ to a specific node $$r \in \mathcal{N}$$. The destination node r is called the root node.

3.1 Network Flow Formulation

The shortest-path problem can be formulated as a network flow problem. Indeed, put a supply of one unit at each nonroot node, and put the appropriate amount of demand at the root (to meet the total supply). The cost on each arc is just the length of the arc. Suppose that we’ve solved this network flow problem. Then the shortest path from a node i to r can be found by simply following the arcs from i to r on the optimal spanning tree. Also, the length of the shortest path is $$y_{r}^{{\ast}}- y_{i}^{{\ast}}$$.

While the network simplex method can be used to solve the shortest-path problem, there are faster algorithms designed especially for it. To describe these algorithms, let us denote the distance from i to r by v i . These distances (or approximations thereof) are called labels in the networks literature. Some algorithms compute these distances systematically in a certain order. These algorithms are called label-setting algorithms. Other algorithms start with an estimate for these labels and then iteratively correct the estimates until the optimal values are found. Such algorithms are called label-correcting algorithms.

Note that if we set y r  ∗  to zero in the network flow solution, then the labels are simply the negative of the optimal dual variables. In the following subsections, we shall describe simple examples of label-setting and label-correcting algorithms.

3.2 A Label-Correcting Algorithm

To describe a label-correcting algorithm, we need to identify a system of equations that characterize the shortest-path distances. First of all, clearly

What can we say about the labels at other nodes, say, node i? Suppose that we select an arc (i, j) that leaves node i. If we were to travel along this arc and then, from node j, travel along the shortest path to r, then the distance to the root would be c ij  + v j . So, from node i, we should select the arc that minimizes these distances. This selection will then give the shortest distance from i to r. That is,

(15.3)

The argument we have just made is called the principle of dynamic programming, and equation (15.3) is called Bellman’s equation. Dynamic programming is a whole subject of its own—we shall only illustrate some of its basic ideas by our study of the shortest-path problem. In the dynamic programming literature, the set of v i ’s viewed as a function defined on the nodes is called the value function (hence the notation).

From Bellman’s equation, it is easy to identify the arcs one would travel on in a shortest-path route to the root. Indeed, these arcs are given by

This set of arcs may contain alternate shortest paths to the root, and so the set is not necessarily a tree. Nonetheless, any path that follows these arcs will get to the root on a shortest-path route.

3.2.1 Method of Successive Approximation

Bellman’s equation is an implicit system of equations for the values v i , $$i \in \mathcal{N}$$. Implicit equations such as this arise frequently and beg to be solved by starting with a guess at the solution, using this guess in the right-hand side, and computing a new guess by evaluating the right-hand side. This approach is called the method of successive approximations. To apply it to the shortest-path problem, we initialize the labels as follows:

Then the updates are computed using Bellman’s equation:

3.2.2 Efficiency

The algorithm stops when an update leaves all the v i ’s unchanged. It turns out that the algorithm is guaranteed to stop in no more than m iterations. To see why, it suffices to note that v i (k) has a very simple description: it is the length of the shortest path from i to r that has k or fewer arcs in the path. (It is not hard to convince yourself with an induction on k that this is correct, but a pedantic proof requires introducing a significant amount of added notation that we wish to avoid.) Hence, the label-correcting algorithm cannot take more than m iterations, since every shortest path can visit each node at most once. Since each iteration involves looking at every arc of the network, it follows that the number of additions/comparisons needed to solve a shortest-path problem using the label-correcting algorithm is about nm.

3.3 A Label-Setting Algorithm

In this section, we describe Dijkstra’s algorithm for solving shortest-path problems. The data structures that are carried from one iteration to the next are a set $$\mathcal{F}$$ of finished nodes and two arrays indexed by the nodes of the graph. The first array, v j , $$j \in \mathcal{N}$$, is just the array of labels. The second array, h i , $$i \in \mathcal{N}$$, indicates the next node to visit from node i in a shortest path. As the algorithm proceeds, the set $$\mathcal{F}$$ contains those nodes for which the shortest path has already been found. This set starts out empty. Each iteration of the algorithm adds one node to it. This is why the algorithm is called a label-setting algorithm, since each iteration sets one label to its optimal value. For finished nodes, the labels are fixed at their optimal values. For each unfinished node, the label has a temporary value, which represents the length of the shortest path from that node to the root, subject to the condition that all intermediate nodes on the path must be finished nodes. At those nodes for which no such path exists, the temporary label is set to infinity (or, in practice, a large positive number).

The algorithm is initialized by setting all the labels to infinity except for the root node, whose label is set to 0. Also, the set of finished nodes is initialized to the empty set. Then, as long as there remain unfinished nodes, the algorithm selects an unfinished node j having the smallest temporary label, adds it to the set of finished nodes, and then updates each unfinished “upstream” neighbor i by setting its label to c ij  + v j if this value is smaller than the current value v i . For each neighbor i whose label gets changed, h i is set to j. The algorithm is summarized in Figure 15.3.

FIGURE 15.3

Dijkstra’s shortest-path algorithm.

4 Upper-Bounded Network Flow Problems

Some real-world network flow problems involve upper bounds on the amount of flow that an arc can handle. There are modified versions of the network simplex method that allow one to handle such upper bounds implicitly, but we shall simply show how to reduce an upper-bounded network flow problem to one without upper bounds.

Let us consider just one arc, (i, j), in a network flow problem. Suppose that there is an upper bound of u ij on the amount of flow that this arc can handle. We can express this bound as an extra constraint:

Introducing a slack variable, t ij , we can rewrite these bound constraints as

If we look at the flow balance constraints and focus our attention on the variables x ij and t ij , we see that they appear in only three constraints: the flow balance constraints for nodes i and j and the upper bound constraint,

If we subtract the last constraint from the second one, we get

Note that we have restored a network structure in the sense that each column again has one + 1 and one − 1 coefficient. To make a network picture, we need to create a new node (corresponding to the third row). Let us call this node k. The network transformation is shown in Figure 15.4.

FIGURE 15.4

Adding a new node, k, to accommodate an arc (i, j) having an upper bound u ij on its flow capacity.

We can use the above transformation to derive optimality conditions for upper-bounded network flow problems. Indeed, let us consider an optimal solution to the transformed problem. Clearly, if x ik is zero, then the corresponding dual slack $$z_{ik} = y_{i} + c_{ij} - y_{k}$$ is nonnegative:

(15.4)

Furthermore, the back-flow x jk must be at the upper bound rate:

Hence, by complementarity, the corresponding dual slack must vanish:

(15.5)

Combining (15.4) with (15.5), we see that

On the other hand, if the flow on arc (i, k) is at the capacity value, then the back-flow on arc (j, k) must vanish. The complementarity conditions then say that

Combining these two statements, we get

Finally, if 0 < x ij  < u ij , then both slack variables vanish, and this implies that

These properties can then be summarized as follows:

(15.6)

While upper-bounded network flow problems have important applications, we admit that our main interest in them is more narrowly focused. It stems from their relation to an important theorem called the Max-Flow Min-Cut Theorem. We shall state and prove this theorem in the next section. The only tool we need to prove this theorem is the above result giving the complementarity conditions when there are upper bounds on arcs. So on with the show.

5 The Maximum-Flow Problem

The subject of this section is the class of problems called maximum-flow problems. These problems form an important topic in the theory of network flows. There are very efficient algorithms for solving them, and they appear as subproblems in many algorithms for the general network flow problem. However, our aim is rather modest. We wish only to expose the reader to one important theorem in this subject, which is called the Max-Flow Min-Cut Theorem.

Before we can state this theorem we need to set up the situation. Suppose that we are given a network $$(\mathcal{N},\mathcal{A})$$, a distinguished node $$s \in \mathcal{N}$$ called the source node, a distinguished node $$t \in \mathcal{N}$$ called the sink node, and upper bounds on the arcs of the network u ij , $$(i,j) \in \mathcal{A}$$. For simplicity, we shall assume that the upper bounds are all finite (although this is not really necessary). The objective is to “push” as much flow from s to t as possible.

To solve this problem, we can convert it to an upper-bounded network flow problem as follows. First, let c ij  = 0 for all arcs $$(i,j) \in \mathcal{A}$$, and let b i  = 0 for every node $$i \in \mathcal{N}$$. Then add one extra arc (t, s) connecting the sink node t back to the source node s, put a negative cost on this arc (say, c ts  = − 1), and let it have infinite capacity $$u_{ts} = \infty$$. Since the only nonzero cost is actually negative, it follows that we shall actually make a profit by letting more and more flow circulate through the network. But the upper bound on the arc capacities limits the amount of flow that it is possible to push through.

In order to state the Max-Flow Min-Cut Theorem, we must define what we mean by a cut. A cut, C, is a set of nodes that contains the source node but does not contain the sink node (see Figure 15.5). The capacity of a cut is defined as

Note that here and elsewhere in this section, the summations are over “original” arcs that satisfy the indicated set membership conditions. That is, they don’t include the arc that we added connecting from t back to s. (If it did, the capacity of every cut would be infinite—which is clearly not our intention.)

FIGURE 15.5

A cut set C for a maximum flow problem.

Flow balance tells us that the total flow along original arcs connecting the cut set C to its complement minus the total flow along original arcs that span these two sets in the opposite direction must equal the amount of flow on the artificial arc (t, s). That is,

(15.7)

We are now ready to state the Max-Flow Min-Cut Theorem.

Theorem 15.1.

The maximum value of x ts equals the minimum value of κ(C).

Proof.

The proof follows the usual sort of pattern common in subjects where there is a sort of duality theory. First of all, we note that it follows from (15.7) that

(15.8)

for every feasible flow and every cut set C. Then all that is required is to exhibit a feasible flow and a cut set for which this inequality is an equality.

Let x ij  ∗ , $$(i,j) \in \mathcal{A}$$, denote the optimal values of the primal variables, and let y i  ∗ , $$i \in \mathcal{N}$$, denote the optimal values of the dual variables. Then the complementarity conditions (15.6) imply that

(15.9)

(15.10)

In particular,

(since u ts  = ∞). Put $${C}^{{\ast}} =\{ k : y_{k}^{{\ast}}\leq y_{s}^{{\ast}}\}$$. Clearly, C  ∗  is a cut.

Consider an arc having its tail in C  ∗  and its head in the complement of C  ∗ . It follows from the definition of C  ∗  that $$y_{i}^{{\ast}}\leq y_{s}^{{\ast}} <y_{j}^{{\ast}}$$. Since c ij is zero, we see from (15.10) that $$x_{ij}^{{\ast}} = u_{ij}$$.

Now consider an original arc having its tail in the complement of C  ∗  and its head in C  ∗  (i.e., bridging the two sets in the opposite direction). It follows then that $$y_{j}^{{\ast}}\leq y_{s}^{{\ast}} <y_{i}^{{\ast}}$$. Hence, we see from (15.9) that $$x_{ij}^{{\ast}} = 0$$.

Combining the observations of the last two paragraphs with (15.7), we see that

In light of (15.8), the proof is complete. □ 

Exercises

15.1

Solve the transportation problem given in (15.1), using (15.2) for the starting tree solution.

15.2

Solve the following linear programming problem:

(Note: there are two greater-than-or-equal-to constraints.)

15.3

Bob, Carol, David, and Alice are stranded on a desert island. Bob and David each would like to give their affection to Carol or to Alice. Food is the currency of trade for this starving foursome. Bob is willing to pay Carol 7 clams if she will accept his affection. David is even more keen and is willing to give Carol 9 clams if she will accept it. Both Bob and David prefer Carol to Alice (sorry Alice). To quantify this preference, David is willing to pay Alice only 2 clams for his affection. Bob is even more averse: he says that Alice would have to pay him for it. In fact, she’d have to pay him 3 clams for his affection. Carol and Alice, being proper young women, will accept affection from one and only one of the two guys. Between the two of them they have decided to share the clams equally between them and hence their objective is simply to maximize the total number of clams they will receive. Formulate this problem as a transportation problem. Solve it.

15.4

Project Scheduling.This problem deals with the creation of a project schedule; specifically, the project of building a house. The project has been divided into a set of jobs. The problem is to schedule the time at which each of these jobs should start and also to predict how long the project will take. Naturally, the objective is to complete the project as quickly as possible (time is money!). Over the duration of the project, some of the jobs can be done concurrently. But, as the following table shows, certain jobs definitely can’t start until others are completed.

One possible schedule is the following:

Job

Duration (weeks)

Must be preceded by

0. Sign contract with buyer

0

–

1. Framing

2

0

2. Roofing

1

1

3. Siding

3

1

4. Windows

2.5

3

5. Plumbing

1.5

3

6. Electrical

2

2,4

7. Inside finishing

4

5,6

8. Outside painting

3

2,4

9. Complete the sale to buyer

0

7,8

With this schedule, the project duration is 21 weeks (the difference between the start times of jobs 9 and 0).

Job

Start time

0. Sign contract with buyer

0

1. Framing

1

2. Roofing

4

3. Siding

6

4. Windows

10

5. Plumbing

9

6. Electrical

13

7. Inside finishing

16

8. Outside painting

14

9. Complete the sale to buyer

21

To model the problem as a linear program, introduce the following decision variables:

(a)

Write an expression for the objective function, which is to minimize the project duration.

(b)

For each job j, write a constraint for each job i that must precede j; the constraint should ensure that job j doesn’t start until job i is finished. These are called precedence constraints.

15.5

Continuation. This problem generalizes the specific example of the previous problem. A project consists of a set of jobs $$\mathcal{J}$$. For each job $$j \in \mathcal{J}$$ there is a certain set $$\mathcal{P}_{j}$$ of other jobs that must be completed before job j can be started. (This is called the set of predecessors of job j.) One of the jobs, say s, is the starting job; it has no predecessors. Another job, say t, is the final (or terminal) job; it is not the predecessor of any other job. The time it will take to do job j is denoted $$d_{j}$$ (the duration of the job).

The problem is to decide what time each job should begin so that no job begins before its predecessors are finished, and the duration of the entire project is minimized. Using the notations introduced above, write out a complete description of this linear programming problem.

15.6

Continuation. Let x ij denote the dual variable corresponding to the precedence constraint that ensures job j doesn’t start until job i finishes.

(a)

Write out the dual to the specific linear program in Problem 15.4.

(b)

Write out the dual to the general linear program in Problem 15.5.

(c)

Describe how the optimal value of the dual variable x ij can be interpreted.

15.7

Continuation. The project scheduling problem can be represented on a directed graph with arc weights as follows. The nodes of the graph correspond to the jobs. The arcs correspond to the precedence relations. That is, if job i must be completed before job j, then there is an arc pointing from node i to node j. The weight on this arc is d i .

(a)

Draw the directed graph associated with the example in Problem 15.4, being sure to label the nodes and write the weights beside the arcs.

(b)

Return to the formulation of the dual from Problem 15.6(a). Give an interpretation of that dual problem in terms of the directed graph drawn in Part (a).

(c)

Explain why there is always an optimal solution to the dual problem in which each variable x ij is either 0 or 1.

(d)

Write out the complementary slackness condition corresponding to dual variable x ₂₆.

(e)

Describe the dual problem in the language of the original project scheduling model.

15.8

Continuation. Here is an algorithm for computing optimal start times t j :

1.

List the jobs so that the predecessors of each job come before it in the list.

2.

Put t ₀ = 0.

3.

Go down the list of jobs and for job j put $$t_{j} =\max \{ t_{i} + d_{i} :\ i\mbox{ is a predecessor of }j\}$$.

(a)

Apply this algorithm to the specific instance from Problem 15.4. What are the start times of each of the jobs? What is the project duration?

(b)

Prove that the solution found in Part (a) is optimal by exhibiting a corresponding dual solution and checking the usual conditions for optimality (Hint: The complementary slackness conditions may help you find a dual solution.).

15.9

Currency Arbitrage. Consider the world’s currency market. Given two currencies, say the Japanese Yen and the US Dollar, there is an exchange rate between them (currently about 110 Yen to the Dollar). It is always true that, if you convert money from one currency to another and then back, you will end up with less than you started with. That is, the product of the exchange rates between any pair of countries is always less than one. However, it sometimes happens that a longer chain of conversions results in a gain. Such a lucky situation is called an arbitrage. One can use a linear programming model to find such situations when they exist.

Consider the following table of exchange rates (which is actual data from the Wall Street Journal on Nov 10, 1996):

param rate:

USD Yen Mark Franc :=

USD . 111.52 1.4987 5.0852

Yen .008966 . .013493 .045593

Mark .6659 73.964 . 3.3823

Franc .1966 21.933 .29507 .

It is not obvious, but the USD → Yen → Mark → USD conversion actually makes $0.002 on each initial dollar.

To look for arbitrage possibilities, one can make a generalized network model, which is a network flow model with the unusual twist that a unit of flow that leaves one node arrives at the next node multiplied by a scale factor—in our example, the currency conversion rate. For us, each currency is represented by a node. There is an arc from each node to every other node. A flow of one unit out of one node becomes a flow of a different magnitude at the head node. For example, one dollar flowing out of the USD node arrives at the Franc node as 5.0852 Francs.

Let x ij denote the flow from node (i.e. currency) i to node j. This flow is measured in the currency of node i.

One node is special; it is the home node, say the US Dollars (USD) node. At all other nodes, there must be flow balance.

(a)

Write down the flow balance constraints at the 3 non-home nodes (Franc, Yen, and Mark).

At the home node, we assume that there is a supply of one unit (to get things started). Furthermore, at this node, flow balance will not be satisfied. Instead one expects a net inflow. If it is possible to make this inflow greater than one, then an arbitrage has been found. Let f be a variable that represents this inflow.

(b)

Using variable f to represent net inflow to the home node, write a flow balance equation for the home node.

Of course, the primal objective is to maximize f.

(c)

Using y i to represent the dual variable associated with the primal constraint for currency i, write down the dual linear program. (Regard the primal variable f as a free variable.)

Now consider the general case, which might involve hundreds of currencies worldwide.

(a)

Write down the model mathematically using x ij for the flow leaving node i heading for node j (measured in the currency of node i), r ij for the exchange rate when converting from currency i to currency j, and f for the net inflow at the home node i  ∗ .

(b)

Write down the dual problem.

(c)

Can you give an interpretation for the dual variables? Hint: It might be helpful to think about the case where r ji  = 1 ∕ r ij for all i, j.

(d)

Comment on the conditions under which your model will be unbounded and/or infeasible.

Notes

The Hitchcock problem was introduced by Hitchcock (1941). Dijkstra’s algorithm was discovered by Dijkstra (1959).

The Max-Flow Min-Cut Theorem was proved independently by Elias et al. (1956), by Ford and Fulkerson (1956) and, in the restricted case where the upper bounds are all integers, by Kotzig (1956). Fulkerson and Dantzig (1955) also proved the Max-Flow Min-Cut Theorem. Their proof uses duality, which is particularly relevant to this chapter.

The classic references for dynamic programming are the books by Bellman (1957) and Howard (1960). Further discussion of label-setting and label-correcting algorithms can be found in the book by Ahuja et al. (1993).

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© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_16

16. Structural Optimization

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

This final chapter on network-type problems deals with finding the best design of a structure to support a specified load at a fixed set of points. The topology of the problem is described by a graph where each node represents a joint in the structure and each arc represents a potential member.¹ We shall formulate this problem as a linear programming problem whose solution determines which of the potential members to include in the structure and how thick each included member must be to handle the load. The optimization criterion is to find a minimal weight structure. As we shall see, the problem bears a striking resemblance to the minimum-cost network flow problem that we studied in Chapter 14.

1 An Example

We begin with an example. Consider the graph shown in Figure 16.1. This graph represents a structure consisting of five joints and eight possible members connecting the joints. The five joints and their coordinates are given as follows: Since joints are analogous to nodes in a network, we shall denote the set of joints by $$\mathcal{N}$$ and denote by m the number of joints. Also, since members are analogous to arcs in network flows, we shall denote the set of them by $$\mathcal{A}$$. For the structure shown in Figure 16.1, the set of members is

Note that we enclosed the pairs of endjoints in braces to emphasize that their order is irrelevant. For example, {2, 3} and {3, 2} refer to one and the same member spanning between joints 2 and 3. In network flows, the graphs we considered were directed graphs. Here, they are undirected. Also, the graphs here are embedded in a d-dimensional Euclidean space (meaning that every node comes with a set of coordinates indicating its location in d-dimensional space). No such embedding was imposed before in our study of network flows, even though real-world network flow problems often possess such an embedding.

Joint

Coordinates

1

(0.0 , 0.0)

2

(6.0 , 0.0)

3

(0.0 , 8.0)

4

(6.0 , 8.0)

5

(3.0 , 12.0)

FIGURE 16.1

Sample topology for a two-dimensional structure.

Following the standard convention of using braces to denote sets, we ought to let x {i, j} denote the force exerted by member {i, j} on its endjoints. But the braces are cumbersome. Hence, we shall write this force simply as x ij , with the understanding that x ij and x ji denote one and the same variable.

We shall assume that a positive force represents tension in the member (i.e., the member is pulling “in” on its two endjoints) and that a negative value represents compression (i.e., the member is pushing “out” on its two endjoints).

If the structure is to be in equilibrium (i.e., not accelerating in some direction), then forces must be balanced at each joint. Of course, we assume that there may be a nonzero external load at each joint (this is the analogue of the external supply/demand in the minimum-cost network flow problem). Hence, for each node i, let b i denote the externally applied load. Note that each b i is a vector whose dimension equals the dimension of the space in which the structure lies. For our example, this dimension is 2. In general, we shall denote the spatial dimension by d.

Force balance imposes a number of constraints on the member forces. For example, the force balance equations for joint 2 can be written as follows:

where b ₂ ¹ and b ₂ ² denote the components of b ₂. Note that the three vectors appearing on the left are unit vectors pointing out from joint 2 along each of the corresponding members.

2 Incidence Matrices

If, for each joint i, we let p i denote its position vector, then the unit vectors pointing along the arcs can be written as follows:

It is important to note that u ji  =  − u ij , since the first vector points from j towards i, whereas the second points from i towards j. In terms of these notations, the force balance equations can be expressed succinctly as

(16.1)

These equations can be written in matrix form as

(16.2)

where x denotes the vector consisting of the member forces, b denotes the vector whose elements are the applied load vectors, and A is a matrix containing the unit vectors pointing along the appropriate arcs. For our example, we have

Note that we have written A as a matrix of 2-vectors by putting “inner” brackets around appropriate pairs of entries. These inner brackets could of course be dropped—they are included simply to show how the constraints match up with (16.1).

In network flows, an incidence matrix is characterized by the property that every column of the matrix has exactly two nonzero entries, one + 1 and one − 1. Here, the matrix A is characterized by the property that, when viewed as a matrix of d-vectors, every column has two nonzero entries that are unit vectors pointing in opposite directions from each other. Generically, matrix A can be written as follows:

where, for each $$\{i,j\} \in \mathcal{A}$$,

and

By analogy with network flows, the matrix A is called an incidence matrix. This definition is a strict generalization of the definition we had before, since, for d = 1, the current notion reduces to the network flows notion. Incidence matrices for network flows enjoy many useful properties. In the following sections, we shall investigate the extent to which these properties carry over to our generalized notion.

3 Stability

Recall that for network flows, the sum of the rows of the incidence matrix vanishes and that if the network is connected, this is the only redundancy. For d > 1, the situation is similar. Clearly, the sum of the rows vanishes. But is this the only redundancy? To answer this question, we need to look for nonzero row vectors y T for which y T A = 0. The set of all such row vectors is a subspace of the set of all row vectors. Our aim is to find a basis for this subspace and, in particular, to identify its dimension. To this end, first write y in component form as $${y}^{T} = [\begin{array}{ccc} y_{1}^{T}\cdots y_{m}^{T} \end{array} ]$$ where each of the entries y i , i = 1, 2, …, m, are d-vectors (transposed to make them into row vectors). Multiplying this row vector against each column of A, we see that y T A = 0 if and only if

(16.3)

There are many choices of y that yield a zero row combination. For example, we can take any vector $$v \in {\mathbb{R}}^{d}$$ and put

Substituting this choice of y i ’s into the left-hand side of (16.3), we get

This set of choices shows that the subspace is at least d-dimensional.

But there are more! They are defined in terms of skew symmetric matrices. A matrix R is called skew symmetric if R T  =  − R. A simple but important property of skew symmetric matrices is that, for every vector ξ,

(16.4)

(see Exercise 16.1). We shall make use of this property shortly. Now, to give more choices of y, let R be a d ×d skew symmetric matrix and put

(recall that p i denotes the position vector for joint i). We need to check (16.3). Substituting this definition of the y’s into the left-hand side in (16.3), we see that

Now substituting in the definition of u ij , we get

Finally, by putting $$\xi = p_{j} - p_{i}$$ and using property (16.4) of skew symmetric matrices, we see that the numerator on the right vanishes. Hence, (16.3) holds.

How many redundancies have we found? For d = 2, there are two independent v-type redundancies and one more R-type. The following two vectors and a matrix can be taken as a basis for these redundancies

For d = 3, there are three independent v-type redundancies and three R-type. Here are three vectors and three matrices that can be taken as a basis for the space of redundancies:

(16.5)

In general, there are d + d(d − 1) ∕ 2 = d(d + 1) ∕ 2 independent redundancies. There could be more. But just as for network flows, where we showed that there is one redundancy if and only if the network is connected, further redundancies represent a defect in the underlying graph structure. In fact, we say that the graph is stable if the rank of the incidence matrix A is exactly md − d(d + 1) ∕ 2, that is, if and only if the above redundancies account for the entire rank deficiency of A.

4 Conservation Laws

Recall that for network flows, not all choices of supplies/demands yield feasible flows. For connected networks, it is necessary and sufficient that the total supply equals the total demand. The situation is similar here. The analogous question is: which external loads give rise to solutions to (16.2)? We have already identified several row vectors y T for which y T A = 0. Clearly, in order to have a solution to (16.2), it is necessary that y T b = 0 for all these row vectors. In particular for every $$v \in {\mathbb{R}}^{d}$$, we see that b must satisfy the following condition:

Bringing the sum inside of the product, we get

Since this must hold for every d-vector v, it follows that

This condition has a simple physical interpretation: the loads, taken in total, must balance.

What about the choices of y T arising from skew symmetric matrices? We shall show that these choices impose the conditions necessary to prevent the structure from spinning around some axis of rotation. To show that this is so, let us first consider the two-dimensional case. For every 2 ×2 skew symmetric matrix R, the load vectors b i , $$i \in \mathcal{N}$$, must satisfy

(16.6)

This expression is a sum of terms of the form (Rp) T b, where p is the position vector of a point and b is a force applied at this point. We claim that (Rp) T b is precisely the torque about the origin created by applying force b at location p. To see this connection between the algebraic expression and its physical interpretation, first decompose p into the product of its length r times a unit vector v pointing in the same direction and rewrite the algebraic expression as

Now, without loss of generality, we may assume that R is the “basis” matrix for the space of skew symmetric matrices,

This matrix has the additional property that its two columns are unit vectors that are orthogonal to each other. That is, R T R = I. Hence,

Furthermore, property (16.4) tells us that Rv is orthogonal to v. Therefore, the product (Rv) T b is the length of the projection of b in the direction of Rv, and so r(Rv) T b is the distance from the origin (of the coordinate system) to p, which is called the moment arm, times the component of the force that is orthogonal to the moment arm in the direction of Rv (see Figure 16.2). This interpretation for each summand in (16.6) shows that it is exactly the torque around the rotation axis passing through the origin of the coordinate system caused by the force b i applied to joint i. In d = 2, there is only one rotation around the origin. This fact corresponds to the fact that the dimension of the space of skew symmetric matrices in two dimensions is 1. Also, stipulating that the total torque about the origin vanishes implies that the total torque around any point other point also vanishes—see Exercise 16.4.

Fig. 16.2

The ith summand in (16.6) is the length of p i times the length of the projection of b i onto the direction given by Rv i . This is precisely the torque around an axis at 0 caused by the force b i applied at joint i.

The situation for d > 2, in particular for d = 3, is slightly more complicated. Algebraically, the complications arise because the basic skew symmetric matrices no longer satisfy R T R = I. Physically, the complications stem from the fact that in two dimensions rotation takes place around a point, whereas in three dimensions it takes place around an axis. We shall explain how to resolve the complications for d = 3. The extension to higher dimensions is straightforward (and perhaps not so important). The basic conclusion that we wish to derive is the same, namely that for basic skew symmetric matrices, the expression (Rp) T b represents the torque generated by applying a force b at point p. Recall that there are just three basic skew symmetric matrices, and they are given by (16.5). To be specific, let us just study the first one:

This matrix can be decomposed into the product of two matrices:

where

The matrix U has the property that R had before, namely,

Such matrices are called unitary. The matrix P is a projection matrix. If we let

and

then we can rewrite (Rp) T b as

Since v is a unit vector and U is unitary, it follows that Uv is a unit vector. Hence, (Uv) T b represents the scalar projection of b onto the direction determined by Uv. Also, it is easy to check that Uv is orthogonal to v. At this point, we can consult Figure 16.3 to see that r is the moment arm for the torque around the third coordinate axis and (Uv) T b is the component of force in the direction of rotation around this axis. Therefore, the product is precisely the torque around this axis. As we know, for d = 3, there are three independent axes of rotation, namely, pitch, roll, and yaw. These axes correspond to the three basis matrices for the space of 3 ×3 skew symmetric matrices (the one we have just studied corresponds to the yaw axis).

Fig. 16.3

The decomposition of (Rp) T b into the product of a moment arm r times the component of b in the direction Uv shows that it is precisely the torque around the third axis.

Finally, we note that (16.6) simply states that the total torque around each axis of rotation must vanish. This means that the forces cannot be chosen to make the system spin.

5 Minimum-Weight Structural Design

For a structure with m nodes, the system of force balance equations (16.2) has md equations. But, as we now know, if the structure is stable, there are exactly d(d + 1) ∕ 2 redundant equations. That is, the rank of A is md − d(d + 1) ∕ 2. Clearly, the structure must contain at least this many members. We say that the structure is a truss if it is stable and has exactly md − d(d + 1) ∕ 2 members. In this case, the force balance equations have a unique solution (assuming, of course, that the total applied force and the total applied torque around each axis vanish). From an optimization point of view, trusses are not interesting because they leave nothing to optimize—one only needs to calculate.

To obtain an interesting optimization problem, we assume that the proposed structure has more members than the minimum required to form a truss. In this setting, we introduce an optimization criterion to pick that solution (whether a truss or otherwise) that minimizes the criterion. For us, we shall attempt to minimize total weight. To keep things simple, we assume that the weight of a member is directly proportional to its volume and that the constant of proportionality (the density of the material) is the same for each member. (These assumptions are purely for notational convenience—a real engineer would certainly include these constants and let them vary from one member to the next). Hence, it suffices to minimize the total volume. The volume of one member, say, {i, j}, is its length $$l_{ij} =\| p_{j} - p_{i}\|$$ times its cross-sectional area. Again, to keep things as simple as possible, we assume that the cross-sectional area must be proportional to the tension/compression carried by the member (members carrying big loads must be “fat”—otherwise they might break). Let’s set the constant of proportionality arbitrarily to one. Then the function that we should minimize is just the sum over all members of l ij  | x ij  | . Hence, our optimization problem can be written as follows:

This problem is not a linear programming problem: the constraints are linear, but the objective function involves the absolute value of each variable. We can, however, convert this problem to a linear programming problem with the following trick. For each $$\{i,j\} \in \mathcal{A}$$, write x ij as the difference between two nonnegative variables:

Think of x ij  +  as the tension part of x ij and x ij  −  as the compression part. The absolute value can then be modeled as the sum of these components

We allow both components to be positive at the same time, but no minimum-weight solution will have any member with both components positive, since if there were such a member, the tension component and the compression component could be decreased simultaneously at the same rate without changing the force balance equations but reducing the weight. This reduction contradicts the minimum-weight assumption.

We can now state the linear programming formulation of the minimum weight structural design problem as follows:

In terms of the incidence matrix, each column must now be written down twice, once as before and once as the negative of before.

6 Anchors Away

So far we have considered structures that are free floating in the sense that even though loads are applied at various joints, we have not assumed that any of the joints are anchored to a large object such as the Earth. This setup is fine for structures intended for a rocket or a space station, but for Earth-bound applications it is generally desired to anchor some joints. It is trivial to modify the formulation we have already given to cover the situation where some of the joints are anchored. Indeed, the d force balance equations associated with an anchored joint are simply dropped as constraints, since the Earth supplies whatever counterbalancing force is needed. Of course, one can consider dropping only some of the d force balance equations associated with a particular joint. In this case, the physical interpretation is quite simple. For example, in two dimensions it simply means that the joint is allowed to roll on a track that is aligned with one of the coordinate directions but is not allowed to move off the track.

If enough “independent” constraints are dropped (at least three in two dimensions and at least six in three dimensions), then there are no longer any limitations on the applied loads—the structure will be sufficiently well anchored so that the Earth will apply whatever forces are needed to prevent the structure from moving. This is the most typical scenario under which these problems are solved. It makes setting up the problem much easier, since one no longer needs to worry about supplying loads that can’t be balanced.

We end this chapter with one realistic example. Suppose the need exists to design a bracket to support a hanging load at a fixed distance from a wall. This bracket will be molded out of plastic, which means that the problem of finding an optimal design belongs to the realm of continuum mechanics. However, we can get an idea of the optimal shape by modeling the problem discretely (don’t tell anyone). That is, we define a lattice of joints as shown in Figure 16.4 and introduce a set of members from which the bracket can be constructed. Each joint has members connecting it to several nearby joints. Figure 16.5 shows the members connected to one specific joint. Each joint in the structure has this connection “topology” with, of course, the understanding that joints close to the boundary do not have any member for which the intended connecting joint does not exist.

FIGURE 16.4

The set of joints used for the discrete approximation to the bracket design problem. The highlighted joints on the left are anchored to the wall, and the highlighted joint on the right must support the hanging load.

The highlighted joints on the left side in Figure 16.4 are the anchored joints, and the highlighted joint on the right side is the joint to which the hanging load is applied (by “hanging,” we mean that the applied load points downward). The optimal solution is shown in Figure 16.6. The thickness of each member is drawn in proportion to the square root of the tension/compression in the member (since if the structure actually exists in three dimensions, the diameter of a member would be proportional to the square root of the cross-sectional area). Also, those members under compression are drawn in dark gray, whereas those under tension are drawn in light gray. Note that the compression members appear to cross the tension members at right angles. These curves are called principle stresses. It is a fundamental result in continuum mechanics that the principle tension stresses cross the principle compression stresses at right angles. We have discovered this result using optimization. Most nonexperts find the solution to this problem to be quite surprising, since it covers such a large area. Yet it is indeed optimal. Also, one can see that the continuum solution should be roughly in the shape of a leaf.

FIGURE 16.5

The members connected to a single interior joint.

FIGURE 16.6

The minimum weight bracket.

Exercises

16.1

Show that a matrix R is skew symmetric if and only if

16.2

Which of the structures shown in Figure 16.7 is stable? (Note: each structure is shown embedded in a convenient coordinate system.)

FIGURE 16.7

Structures for Exercises 16.2 and 16.3.

16.3

Which of the structures shown in Figure 16.7 is a truss?

16.4

Assuming that the total applied force vanishes, show that total torque is translation invariant. That is, for any vector $$\xi \in {\mathbb{R}}^{d}$$,

16.5

In 3-dimensions there are 5 regular (Platonic) solids. They are shown in Figure 16.8 and have the following number of vertices and edges:

vertices

edges

tetrahedron

4

6

cube

8

12

octahedron

6

12

dodecahedron

20

30

icosahedron

12

30

FIGURE 16.8

The five regular solids.

If one were to construct pin-jointed wire-frame models of these solids, which ones would be stable?

Notes

Structural optimization has its roots in Michell (1904). The first paper in which truss design was formulated as a linear programming problem is Dorn et al. (1964). A few general references on the subject include Hemp (1973), Rozvany (1989), Bendsøe et al. (1994), and Recski (1989).

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Footnotes

1

Civil engineers refer to beams as members.

Part 3
Interior-Point Methods

There is, I believe, in every disposition a tendency to some particular evil—a natural defect, which not even the best education can overcome.—J. Austen

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_17

17. The Central Path

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In this chapter, we begin our study of an alternative to the simplex method for solving linear programming problems. The algorithm we are going to introduce is called a path-following method. It belongs to a class of methods called interior-point methods. The path-following method seems to be the simplest and most natural of all the methods in this class, so in this book we focus primarily on it. Before we can introduce this method, we must define the path that appears in the name of the method. This path is called the central path and is the subject of this chapter. Before discussing the central path, we must lay some groundwork by analyzing a nonlinear problem, called the barrier problem, associated with the linear programming problem that we wish to solve.

Warning: Nonstandard Notation Ahead

Starting with this chapter, given a lower-case letter denoting a vector quantity, we shall use the upper-case form of the same letter to denote the diagonal matrix whose diagonal entries are those of the corresponding vector. For example,

This notation is nonstandard in mathematics at large, but has achieved a certain amount of acceptance in the interior-point-methods community.

1 The Barrier Problem

In this chapter, we consider the linear programming problem expressed, as usual, with inequality constraints and nonnegative variables:

The corresponding dual problem is

As usual, we add slack variables to convert both problems to equality form:

(17.1)

and

Given a constrained maximization problem where some of the constraints are inequalities (such as our primal linear programming problem), one can consider replacing any inequality constraint with an extra term in the objective function. For example, in (17.1) we could remove the constraint that a specific variable, say, x j , is nonnegative by adding to the objective function a term that is negative infinity when x j is negative and is zero otherwise. This reformulation doesn’t seem to be particularly helpful, since this new objective function has an abrupt discontinuity that, for example, prevents us from using calculus to study it. However, suppose we replace this discontinuous function with another function that is negative infinity when x j is negative but is finite for x j positive and approaches negative infinity as x j approaches zero. In some sense this smooths out the discontinuity and perhaps improves our ability to apply calculus to its study. The simplest such function is the logarithm. Hence, for each variable, we introduce a new term in the objective function that is just a constant times the logarithm of the variable:

(17.2)

This problem, while not equivalent to our original problem, seems not too different either. In fact, as the parameter μ, which we assume to be positive, gets small, it appears that (17.2) becomes a better and better stand-in for (17.1). Problem (17.2) is called the barrier problem associated with (17.1). Note that it is not really one problem, but rather a whole family of problems indexed by the parameter μ. Each of these problems is a nonlinear programming problem because the objective function is nonlinear. This nonlinear objective function is called a barrier function or, more specifically, a logarithmic barrier function.

It is instructive to have in mind a geometric picture of the barrier function. Recall that, for problems expressed in standard form, the set of feasible solutions is a polyhedron with each face being characterized by the property that one of the variables is zero. Hence, the barrier function is minus infinity on each face of the polyhedron. Furthermore, it is finite in the interior of the polyhedron, and it approaches minus infinity as the boundary is approached. Figure 17.1 shows some level sets for the barrier function for a specific problem and a few different choices of μ. Notice that, for each μ, the maximum is attained at an interior point, and as μ gets closer to zero this interior point moves closer to the optimal solution of the original linear programming problem (which is at the top vertex). Viewed as a function of μ, the set of optimal solutions to the barrier problems forms a path through the interior of the polyhedron of feasible solutions. This path is called the central path. Our aim is to study this central path. To this end, we need to develop some machinery, referred to as Lagrange multipliers.

FIGURE 17.1

Parts (a) through (c) show level sets of the barrier function for three values of μ. For each value of μ, four level sets are shown. The maximum value of the barrier function is attained inside the innermost level set. The drawing in part (d) shows the central path.

2 Lagrange Multipliers

We wish to discuss briefly the general problem of maximizing a function subject to one or more equality constraints. Here, the functions are permitted to be nonlinear, but are assumed to be smooth, say, twice differentiable.

For the moment, suppose that there is a single constraint equation so that the problem can be formally stated as

In this case, the geometry behind the problem is compelling (see Figure 17.2). The gradient of f, denoted $$\nabla f$$, is a vector that points in the direction of most rapid increase of f. For unconstrained optimization, we would simply set this vector equal to zero to determine the so-called critical points of f, and the maximum, if it exists, would have to be included in this set. However, given the constraint, g(x) = 0, it is no longer correct to look at points where the gradient vanishes. Instead, the gradient must be orthogonal to the set of feasible solutions {x : g(x) = 0}. Of course, at each point x in the feasible set, $$\nabla g(x)$$, is a vector that is orthogonal to the feasible set at this point x. Hence, our new requirement for a point x  ∗  to be a critical point is that it is feasible and that $$\nabla f({x}^{{\ast}})$$ be proportional to $$\nabla g({x}^{{\ast}})$$. Writing this out as a system of equations, we have

Here, y is the proportionality constant. Note that it can be any real number, either positive, negative, or zero. This proportionality constant is called a Lagrange multiplier.

FIGURE 17.2

The concentric rings illustrate a few level sets of f. Clearly, at the optimal solution, x  ∗ , the gradient must be perpendicular to the feasible set.

Now consider several constraints:

In this case, the feasible region is the intersection of m hypersurfaces (see Figure 17.3). The space orthogonal to the feasible set at a point x is no longer a one-dimensional set determined by a single gradient, but is instead a higher-dimensional space (typically m), given by the span of the gradients. Hence, we require that $$\nabla f({x}^{{\ast}})$$ lie in this span. This yields the following set of equations for a critical point:

(17.3)

FIGURE 17.3

The feasible set is the curve formed by the intersection of g ₁ = 0 and g ₂ = 0. The point x  ∗  is optimal, since the gradient of f at that point is perpendicular to the feasible set.

The derivation of these equations has been entirely geometric, but there is also a simple algebraic formalism that yields the same equations. The idea is to introduce the so-called Lagrangian function

and to look for its critical points over both x and y. Since this is now an unconstrained optimization problem, the critical points are determined by simply setting all the first derivatives to zero:

Writing these equations in vector notation, we see that they are exactly the same as those derived using the geometric approach. These equations are usually referred to as the first-order optimality conditions.

Determining whether a solution to the first-order optimality conditions is indeed a global maximum as desired can be difficult. However, if the constraints are all linear, the first step (which is often sufficient) is to look at the matrix of second derivatives:

This matrix is called the Hessian of f at x. We have

Theorem 17.1.

If the constraints are linear, a critical point x ∗ is a local maximum if

(17.4)

for each ξ ≠ 0 satisfying

(17.5)

Proof.

We start with the two-term Taylor series expansion of f about x  ∗ :

The vector ξ represents a displacement from the current point x  ∗ . The only displacements that are relevant are those that lie in the feasible set. Hence, let ξ be a direction vector satisfying (17.5). From (17.3) and (17.5), we see that $$\nabla f{({x}^{{\ast}})}^{T}\xi = 0$$, and so

Employing (17.4) finishes the proof. □ 

It is worth remarking that if (17.4) is satisfied not just at x  ∗  but at all x, then x  ∗  is a unique global maximum.

In the next section, we shall use Lagrange multipliers to study the central path defined by the barrier problem.

3 Lagrange Multipliers Applied to the Barrier Problem

In this section, we shall use the machinery of Lagrange multipliers to study the solution to the barrier problem. In particular, we will show that (subject to some mild assumptions) for each value of the barrier parameter μ, there is a unique solution to the barrier problem. We will also show that as μ tends to zero, the solution to the barrier problem tends to the solution to the original linear programming problem. In the course of our study, we will stumble naturally upon the central path for the dual problem. Taken together, the equations defining the primal and the dual central paths play an important role, and so we will introduce the notion of a primal–dual central path.

We begin by recalling the barrier problem:

This is an equality-constrained optimization problem, and so it is a problem to which we can apply the Lagrange multiplier tools developed in the previous section. The Lagrangian for this problem is

Taking derivatives with respect to each variable and setting them to zero, we get the first-order optimality conditions:

Writing these equations in matrix form, we get

Here, as warned at the beginning of the chapter, X denotes the diagonal matrix whose diagonal entries are the components of x, and similarly for W. Also, recall that we use e to denote the vector of all ones.

Introducing an extra vector defined as $$z =\mu {X}^{-1}e$$, we can rewrite the first-order optimality conditions like this:

Finally, if we multiply the third equation through by X and the fourth equation by W, we arrive at a primal–dual symmetric form for writing these equations:

(17.6)

Note that the first equation is the equality constraint that appears in the primal problem, while the second equation is the equality constraint for the dual problem. Furthermore, writing the third and fourth equations out componentwise,

we see that they are closely related to our old friend: complementarity. In fact, if we set μ to zero, then they are exactly the usual complementarity conditions that must be satisfied at optimality. For this reason, we call these last two equations the μ-complementarity conditions.

The first-order optimality conditions, as written in (17.6), give us 2n + 2m equations in 2n + 2m unknowns. If these equations were linear, they could be solved using Gaussian elimination, and the entire subject of linear programming would be no more difficult than solving systems of linear equations. But alas, they are nonlinear—but just barely. The only nonlinear expressions in these equations are simple multiplications such as x j z j . This is about the closest to being linear that one could imagine. Yet, it is this nonlinearity that makes the subject of linear programming nontrivial.

We must ask both whether a solution to (17.6) exists and if so is it unique. We address these questions in reverse order.

4 Second-Order Information

To show that the solution, if it is exists, must be unique, we use second-order information on the barrier function:

(17.7)

The first derivatives are

and the pure second derivatives are

All the mixed second derivatives vanish. Therefore, the Hessian is a diagonal matrix with strictly negative entries. Hence, by Theorem 17.1, there can be at most one critical point and, if it exists, it is a global maximum.

5 Existence

So, does a solution to the barrier problem always exist? It might not. Consider, for example, the following trivial optimization problem on the nonnegative half-line:

For this problem, the barrier function is

which doesn’t have a maximum (or, less precisely, the maximum is infinity which is attained at x = ∞). However, such examples are rare. For example, consider modifying the objective function in this example to make x = 0 the unique optimal solution:

In this case, the barrier function is

which is a function whose maximum is attained at x = μ.

In general, we have the following result:

Theorem 17.2.

There exists a solution to the barrier problem if and only if both the primal and the dual feasible regions have nonempty interior.

Proof.

The “only if” part is trivial and less important to us. Therefore, we only prove the “if” part. To this end, suppose that both the primal and the dual feasible regions have nonempty interior. This means that there exists a primal feasible point $$(\bar{x},\bar{w})$$ with $$\bar{x}> 0$$ and $$\bar{w}> 0$$ and there exists a dual feasible point $$(\bar{y},\bar{z})$$ with $$\bar{y}> 0$$ and $$\bar{z}> 0$$.¹ Now, given any primal feasible point (x, w), consider the expression $$\bar{{z}}^{T}x +\bar{ {y}}^{T}w$$. Replacing the primal and dual slack variables with their definitions, we can rewrite this expression as follows:

Solving this equation for the primal objective function c T x, we get that

Therefore, the barrier function f defined in equation (17.7) can be written as follows:

Note that the last term is just a constant. Also, each summand in the two sums is a function of just one variable. These functions all have the following general form:

where a > 0. Such functions have a unique maximum (at μ ∕ a) and tend to − ∞ as ξ tends to ∞. From these observations, it is easy to see that, for every constant c, the set

is bounded.

Put

and let

Clearly, $$\bar{P}$$ is nonempty, since it contains $$(\bar{x},\bar{w})$$. From the discussion above, we see that $$\bar{P}$$ is a bounded set.

This set is also closed. To see this, note that it is the intersection of three sets,

The first two of these sets are obviously closed. The third set is closed because it is the inverse image of a closed set, $$[\bar{f},\infty ]$$, under a continuous mapping f. Finally, the intersection of three closed sets is closed.

In Euclidean spaces, a closed bounded set is called compact. A well-known theorem from real analysis about compact sets is that a continuous function on a nonempty compact set attains its maximum. This means that there exists a point in the compact set at which the function hits its maximum. Applying this theorem to f on $$\bar{P}$$, we see that f does indeed attain its maximum on $$\bar{P}$$, and this implies it attains its maximum on all of {(x, w) : x > 0, w > 0}, since $$\bar{P}$$ was by definition that part of this domain on which f takes large values (bigger than $$\bar{f}$$, anyway). This completes the proof. □ 

We summarize our main result in the following corollary:

Corollary 17.1.

If a primal feasible set (or, for that matter, its dual) has a nonempty interior and is bounded, then for each μ > 0 there exists a unique solution

to (17.6) .

Proof.

Follows immediately from the previous theorem and Exercise 10.7. □ 

The path $$\{(x_{\mu },w_{\mu },y_{\mu },z_{\mu }) :\mu> 0\}$$ is called the primal–dual central path. It plays a fundamental role in interior-point methods for linear programming. In the next chapter, we define the simplest interior-point method. It is an iterative procedure that at each iteration attempts to move toward a point on the central path that is closer to optimality than the current point.

Exercises

17.1

Compute and graph the central trajectory for the following problem:

Hint: The primal and dual problems are the same—exploit this symmetry.

17.2

Let θ be a fixed parameter, $$0 \leq \theta \leq \frac{\pi } {2}$$, and consider the following problem:

Compute an explicit formula for the central path $$(x_{\mu },w_{\mu },y_{\mu },z_{\mu })$$, and evaluate $$\lim _{\mu \rightarrow \infty }x_{\mu }$$ and $$\lim _{\mu \rightarrow 0}x_{\mu }.$$

17.3

Suppose that $$\{x : Ax \leq b,x \geq 0\}$$ is bounded. Let $$r \in {\mathbb{R}}^{n}$$ and $$s \in {\mathbb{R}}^{m}$$ be vectors with positive elements. By studying an appropriate barrier function, show that there exists a unique solution to the following nonlinear system:

17.4

Consider the linear programming problem in equality form:

(17.8)

where each a j is a vector in $${\mathbb{R}}^{m}$$, as is b. Consider the change of variables,

and the associated maximization problem:

(17.9)

(note that the nonnegativity constraints are no longer needed). Let V denote the set of basic feasible solutions to (17.8), and let W denote the set of points $$(\xi _{1}^{2},\xi _{2}^{2},\ldots ,\xi _{n}^{2})$$ in $${\mathbb{R}}^{n}$$ for which $$(\xi _{1},\xi _{2},\ldots ,\xi _{n})$$ is a solution to the first-order optimality conditions for (17.9). Show that V ⊂ W. What does this say about the possibility of using (17.9) as a vehicle to solve (17.8)?

Notes

Research into interior-point methods has its roots in the work of Fiacco and McCormick (1968). Interest in these methods exploded after the appearance of the seminal paper Karmarkar (1984). Karmarkar’s paper uses clever ideas from projective geometry. It doesn’t mention anything about central paths, which have become fundamental to the theory of interior-point methods. The discovery that Karmarkar’s algorithm has connections with the primal–dual central path introduced in this chapter can be traced to Megiddo (1989). The notion of central points can be traced to pre-Karmarkar times with the work of Huard (1967). D.A. Bayer and J.C. Lagarias, in a pair of papers (Bayer and Lagarias 1989a,b), give an in-depth study of the central path.

Deriving optimality conditions and giving conditions under which they are necessary and sufficient to guarantee optimality is one of the main goals of nonlinear programming. Standard texts on this subject include the books by Luenberger (1984), Bertsekas (1995), and Nash and Sofer (1996).

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Footnotes

1

Recall that we write ξ > 0 to mean that ξ j  > 0 for all j.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_18

18. A Path-Following Method

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In this chapter, we define an interior-point method for linear programming that is called a path-following method. Recall that for the simplex method we required a two-phase solution procedure. The path-following method is a one-phase method. This means that the method can begin from a point that is neither primal nor dual feasible and it will proceed from there directly to the optimal solution. Hence, we start with an arbitrary choice of strictly positive values for all the primal and dual variables, i.e., (x, w, y, z) > 0, and then iteratively update these values as follows:

1.

Estimate an appropriate value for μ (i.e., smaller than the “current” value but not too small).

2.

Compute step directions (Δx, Δw, Δy, Δz) pointing approximately at the point $$(x_{\mu },w_{\mu },y_{\mu },z_{\mu })$$ on the central path.

3.

Compute a step length parameter θ such that the new point

continues to have strictly positive components.

4.

Replace (x, w, y, z) with the new solution $$(\tilde{x},\tilde{w},\tilde{y},\tilde{z})$$.

To fully define the path-following method, it suffices to make each of these four steps precise. Since the second step is in some sense the most fundamental, we start by describing that one after which we turn our attention to the others.

1 Computing Step Directions

Our aim is to find (Δx, Δw, Δy, Δz) such that the new point $$(x + \Delta x,w + \Delta w,y + \Delta y,z + \Delta z)$$ lies approximately on the primal–dual central path at the point $$(x_{\mu },w_{\mu },y_{\mu },z_{\mu })$$. Recalling the defining equations for this point on the central path,

we see that the new point $$(x + \Delta x,w + \Delta w,y + \Delta y,z + \Delta z)$$, if it were to lie exactly on the central path at μ, would be defined by

Thinking of (x, w, y, z) as data and (Δx, Δw, Δy, Δz) as unknowns, we rewrite these equations with the unknowns on the left and the data on the right:

Note that we have introduced abbreviated notations, ρ and σ, for the first two right-hand sides. These two vectors represent the primal infeasibility and the dual infeasibility, respectively.

Now, just as before, these equations form a system of nonlinear equations (this time for the “delta” variables). We want to have a linear system, so at this point we simply drop the nonlinear terms to arrive at the following linear system:

(18.1)

(18.2)

(18.3)

(18.4)

This system of equations is a linear system of 2n + 2m equations in 2n + 2m unknowns. We will show later that this system is nonsingular (under the mild assumption that A has full rank) and therefore that it has a unique solution that defines the step directions for the path-following method. Chapters 19 and 20 are devoted to studying methods for efficiently solving systems of this type.

If the business of dropping the nonlinear “delta” terms strikes you as bold, let us remark that this is the most common approach to solving nonlinear systems of equations. The method is called Newton’s method. It is described briefly in the next section.

2 Newton’s Method

Given a function

from ℝ N into ℝ N , a common problem is to find a point ξ  ∗  ∈ ℝ N for which F(ξ  ∗ ) = 0. Such a point is called a root of F. Newton’s method is an iterative method for solving this problem. One step of the method is defined as follows. Given any point ξ ∈ ℝ N , the goal is to find a step directionΔξ for which $$F(\xi +\Delta \xi ) = 0$$. Of course, for a nonlinear F it is not possible to find such a step direction. Hence, it is approximated by the first two terms of its Taylor’s series expansion,

where

The approximation is linear in Δξ. Hence, equating it to zero gives a linear system to solve for the step direction:

Given Δξ, Newton’s method updates the current solution ξ by replacing it with ξ + Δξ. The process continues until the current solution is approximately a root (i.e., F(ξ) ≈ 0). Simple one-dimensional examples given in every elementary calculus text illustrate that this method works well, when it works, but it can fail if F is not well behaved and the initial point is too far from a solution.

Let’s return now to the problem of finding a point on the central path. Letting

and

we see that the set of equations defining $$(x_{\mu },w_{\mu },y_{\mu },z_{\mu })$$ is a root of F. The matrix of derivatives of F is given by

Noting that

it is easy to see that the Newton direction coincides with the direction obtained by solving equations (18.1)–(18.4).

3 Estimating an Appropriate Value for the Barrier Parameter

We need to say how to pick μ. If μ is chosen to be too large, then the sequence could converge to the analytic center of the feasible set, which is not our intention. If, on the other hand, μ is chosen to be too small, then the sequence could stray too far from the central path and the algorithm could jam into the boundary of the feasible set at a place that is suboptimal. The trick is to find a reasonable compromise between these two extremes. To do this, we first figure out a value that represents, in some sense, the current value of μ and we then choose something smaller than that, say a fixed fraction of it.

We are given a point (x, w, y, z) that is almost certainly off the central path. If it were on the central path, then there are several formulas by which we could recover the corresponding value of μ. For example, we could just compute z j x j for any fixed index j. Or we could compute y i w i for any fixed i. Or, perverse as it may seem, we could average all these values:

(18.5)

This formula gives us exactly the value of μ whenever it is known that (x, w, y, z) lies on the central path. The key point here then is that we will use this formula to produce an estimate for μ even when the current solution (x, w, y, z) does not lie on the central path. Of course, the algorithm needs a value of μ that represents a point closer to optimality than the current solution. Hence, the algorithm takes this “par” value and reduces it by a certain fraction:

where δ is a number between zero and one. In practice, one finds that setting δ to approximately 1/10 works quite well, but for the sake of discussion we will always leave it as a parameter.

4 Choosing the Step Length Parameter

The step directions, which were determined using Newton’s method, were determined under the assumption that the step length parameter θ would be equal to one (i.e., $$\tilde{x} = x + \Delta x$$, etc.). But taking such a step might cause the new solution to violate the property that every component of all the primal and the dual variables must remain positive. Hence, we may need to use a smaller value for θ. We need to guarantee, for example, that

Moving the Δx j term to the other side and then dividing through by θ and x j , both of which are positive, we see that θ must satisfy

Of course, a similar inequality must be satisfied for the w, y, and z variables too. Putting it all together, the largest value of θ would be given by

where we have abused notation slightly by using the max ij to denote the maximum of all the ratios in the indicated set. However, this choice of θ will not guarantee strict inequality, so we introduce a parameter r, which is a number close to but strictly less than one, and we set¹

(18.6)

This formula may look messy, and no one should actually do it by hand, but it is trivial to program a computer to do it. Such a subroutine will be really fast (requiring only on the order of 2n + 2m operations).

A summary of the algorithm is shown in Figure 18.1. In the next section, we investigate whether this algorithm actually converges to an optimal solution.

FIGURE 18.1

The path-following method.

5 Convergence Analysis

In this section, we investigate the convergence properties of the path-following algorithm. Recall that the simplex method is a finite algorithm (assuming that steps are taken to guard against cycling). For interior-point methods, the situation is different. Every solution produced has all variables strictly positive. Yet for a solution to be optimal generally requires many variables to vanish. This vanishing can only happen “in the limit.” This raises questions, the most fundamental of which are these: does the sequence of solutions produced by the path-following method converge? If so, is the limit optimal? How fast is the convergence? In particular, if we set “optimality tolerances,” how many iterations will it take to achieve these tolerances? We will address these questions in this section.

In this section, we will need to measure the size of various vectors. There are many choices. For example, for each 1 ≤ p < ∞, we can define the so-called p-norm of a vector x as

The limit as p tends to infinity is also well defined, and it simplifies to the so-called sup-norm:

5.1 Measures of Progress

Recall from duality theory that there are three criteria that must be met in order that a primal–dual solution be optimal:

1.

Primal feasibility,

2.

Dual feasibility, and

3.

Complementarity.

For each of these criteria, we introduce a measure of the extent to which they fail to be met.

For the primal feasibility criterion, we use the 1-norm of the primal infeasibility vector

For the dual feasibility criterion, we use the 1-norm of the dual infeasibility vector

For complementarity, we use

5.2 Progress in One Iteration

For the analysis in the section, we prefer to modify the algorithm slightly by having it take shorter steps than specified before. Indeed, we let

(18.7)

Note that the only change has been to replace the negative ratios with the absolute value of the same ratios. Since the maximum of the absolute values can be larger than the maximum of the ratios themselves, this formula produces a smaller value for θ. In this section, let x, y, etc., denote quantities from one iteration of the algorithm, and put a tilde on the same letters to denote the same quantity at the next iteration of the algorithm. Hence,

Now let’s compute some of the other quantities. We begin with the primal infeasibility:

But $$b - Ax - w$$ equals the primal infeasibility ρ (by definition) and AΔx + Δw also equals ρ, since this is precisely the first equation in the system that defines the “delta” variables. Hence,

(18.8)

Similarly,

(18.9)

Since θ is a number between zero and one, it follows that each iteration produces a decrease in both the primal and the dual infeasibility and that this decrease is better the closer θ is to one.

The analysis of the complementarity is a little more complicated (after all, this is the part of the system where the linearization took place):

We need to analyze each of the θ terms separately. From (18.3), we see that

Similarly, from (18.4), we have

Finally, (18.1) and (18.2) imply that

Substituting these expressions into the last expression for $$\tilde{\gamma }$$, we get

At this point, we recognize that $${z}^{T}x + {y}^{T}w =\gamma$$ and that $$\mu (n + m) =\delta \gamma$$. Hence,

We must now abandon equalities and work with estimates. Our favorite tool for estimation is the following inequality:

This inequality is the trivial case of Hölder’s inequality. From Hölder’s inequality, we see that

Hence,

Next, we use the specific choice of step length θ to get a bound on $$\|\theta \Delta y\|_{\infty }$$ and $$\|\theta \Delta x\|_{\infty }$$. Indeed, (18.7) implies that

Hence,

Similarly,

If we now assume that, along the sequence of points x and y visited by the algorithm, $$\|x\|_{\infty }$$ and $$\|y\|_{\infty }$$ are bounded by a large real number M, then we can estimate the new complementarity as

(18.10)

5.3 Stopping Rule

Let ε > 0 be a small positive tolerance, and let M < ∞ be a large finite tolerance. If $$\|x\|_{\infty }$$ gets larger than M, then we stop and declare the problem primal unbounded. If $$\|y\|_{\infty }$$ gets larger than M, then we stop and declare the problem dual unbounded. Finally, if $$\|\rho \|_{1} <\epsilon$$, $$\|\sigma \|_{1} <\epsilon$$, and γ < ε, then we stop and declare the current solution to be optimal (at least within this small tolerance).

Since γ is a measure of complementarity and complementarity is related to the duality gap, one would expect that a small value of γ should translate into a small duality gap. This turns out to be true. Indeed, from the definitions of γ, σ, and ρ, we can write

At this point, we use Hölder’s inequality to bound some of these terms to get an estimate on the duality gap:

Now, if γ, $$\|\sigma \|_{1}$$, and $$\|\rho \|_{1}$$ are all small (and $$\|x\|_{\infty }$$ and $$\|y\|_{\infty }$$ are not too big), then the duality gap will be small. This estimate shows that one shouldn’t expect the duality gap to get small until the primal and the dual are very nearly feasible. Actual implementations confirm this expectation.

5.4 Progress Over Several Iterations

Now let ρ (k), σ (k), γ (k), θ (k), etc., denote the values of these quantities at the kth iteration. We have the following result about the overall performance of the algorithm:

Theorem 18.1.

Suppose there is a real number t > 0, a real number M < ∞, and an integer K such that for all k ≤ K,

Then there exists a constant $$\bar{M} < \infty $$ such that

for all k ≤ K where

Proof.

From (18.8) and the bound on θ (k), it follows that

Similarly, from (18.9), it follows that

As usual, γ (k) is harder to estimate. From (18.10) and the previous two estimates, we see that

(18.11)

where $$\tilde{M} = M\left (\|{\rho }^{(0)}\|_{1} +\|{\sigma }^{(0)}\|_{1}\right )$$. Since an analogous inequality relates γ (k − 1) to γ (k − 2), we can substitute this analogous inequality into (18.11) to get

Continuing in this manner, we see that

Now we sum the bracketed partial sum of a geometric series to get

Recalling that $$\tilde{t} = t(1-\delta )$$ and dropping the second term in the numerator, we get

Putting this all together, we see that

Denoting the parenthesized expression by $$\bar{M}$$ completes the proof. □ 

Theorem 18.1 is only a partial convergence result because it depends on the assumption that the step lengths remain bounded away from zero. To show that the step lengths do indeed have this property requires that the algorithm be modified and that the starting point be carefully selected. The details are rather technical and hence omitted (see the Notes at the end of the chapter for references).

Also, before we leave this topic, note that the primal and dual infeasibilities go down by a factor of 1 − t at each iteration, whereas the duality gap goes down by a smaller amount $$1 -\tilde{ t}$$. The fact that the duality gap converges more slowly that the infeasibilities is also readily observed in practice.

Exercises

18.1

Starting from (x, w, y, z) = (e, e, e, e), and using $$\delta = 1/10$$, and $$r = 9/10$$, compute (x, w, y, z) after one step of the path-following method for the problem given in

(a)

Exercise 2.3.

(b)

Exercise 2.4.

(c)

Exercise 2.5.

(d)

Exercise 2.10.

18.2

Let $$\{(x_{\mu },w_{\mu },y_{\mu },z_{\mu }) :\mu \geq 0\}$$ denote the central trajectory. Show that

Hint: look at (18.5).

18.3

Consider a linear programming problem whose feasible region is bounded and has nonempty interior. Use the result of Exercise 18.2 to show that the dual problem’s feasible set is unbounded.

18.4

Scale invariance. Consider a linear program and its dual:

Let R and S be two given diagonal matrices having positive entries along their diagonals. Consider the scaled reformulation of the original problem and its dual:

Let $$({x}^{k},{w}^{k},{y}^{k},{z}^{k})$$ denote the sequence of solutions generated by the primal–dual interior-point method applied to (P)–(D). Similarly, let $$(\bar{{x}}^{k},\bar{{w}}^{k},\bar{{y}}^{k},\bar{{z}}^{k})$$ denote the sequence of solutions generated by the primal–dual interior-point method applied to $$(\bar{P})$$–$$(\bar{D})$$. Suppose that we have the following relations among the starting points:

Show that these relations then persist. That is, for each k ≥ 1,

18.5

Homotopy method. Let $$\bar{x}$$, $$\bar{y}$$, $$\bar{z}$$, and $$\bar{w}$$ be given componentwise positive “initial” values for x, y, z, and w, respectively. Let t be a parameter between 0 and 1. Consider the following nonlinear system:

(18.12)

(a)

Use Exercise 17.3 to show that this nonlinear system has a unique solution for each 0 ≤ t < 1. Denote it by (x(t), y(t), z(t), w(t)).

(b)

Show that $$(x(0),y(0),z(0),w(0)) = (\bar{x},\bar{y},\bar{z},\bar{w})$$.

(c)

Assuming that the limit

exists, show that it solves the standard-form linear programming problem.

(d)

The family of solutions (x(t), y(t), z(t), w(t)), 0 ≤ t < 1, describes a curve in “primal–dual” space. Show that the tangent to this curve at t = 0 coincides with the path-following step direction at $$(\bar{x},\bar{y},\bar{z},\bar{w})$$ computed with μ = 0; that is,

where (Δx, Δy, Δz, Δw) is the solution to (18.1)–(18.4).

18.6

Higher-order methods. The previous exercise shows that the path-follow-ing step direction can be thought of as the direction one gets by approximating a homotopy path with its tangent line:

By using more terms of the Taylor’s series expansion, one can get a better approximation:

(a)

Differentiating the equations in (18.12) twice, derive a linear system for $$({d}^{2}x/d{t}^{2}(0),{d}^{2}y/d{t}^{2}(0),{d}^{2}z/d{t}^{2}(0),{d}^{2}w/d{t}^{2}(0))$$.

(b)

Can the same technique be applied to derive linear systems for the higher-order derivatives?

18.7

Linear Complementarity Problem. Given a k ×k matrix M and a k-vector q, a vector x is said to solve the linear complementarity problem if

(note that the first equation can be taken as the definition of z).

(a)

Show that the optimality conditions for linear programming can be expressed as a linear complementarity problem with

(b)

The path-following method introduced in this chapter can be extended to cover linear complementarity problems. The main step in the derivation is to replace the complementarity condition XZe = 0 with a μ-complementarity condition XZe = μe and then to use Newton’s method to derive step directions Δx and Δz. Carry out this procedure and indicate the system of equations that define Δx and Δz.

(c)

Give conditions under which the system derived above is guaranteed to have a unique solution.

(d)

Write down the steps of the path-following method for the linear complementarity problem.

(e)

Study the convergence of this algorithm by adapting the analysis given in Section 18.5.

18.8

Consider again the L ¹-regression problem:

Complete the following steps to derive the step direction vector Δx associated with the primal–dual affine-scaling method for solving this problem.

(a)

Show that the L ¹-regression problem is equivalent to the following linear programming problem:

(18.13)

(b)

Write down the dual of (18.13).

(c)

Add slack and/or surplus variables as necessary to reformulate the dual so that all inequalities are simple nonnegativities of variables.

(d)

Identify all primal–dual pairs of complementary variables.

(e)

Write down the nonlinear system of equations consisting of: (1) the primal equality constraints, (2) the dual equality constraints, (3) all complementarity conditions (using μ = 0 since we are looking for an affine-scaling algorithm).

(f)

Apply Newton’s method to the nonlinear system to obtain a linear system for step directions for all of the primal and dual variables.

(g)

We may assume without loss of generality that both the initial primal solution and the initial dual solution are feasible. Explain why.

(h)

The linear system derived above is a 6 ×6 block matrix system. But it is easy to solve most of it by hand. First eliminate those step directions associated with the nonnegative variables to arrive at a 2 ×2 block matrix system.

(i)

Next, solve the 2 ×2 system. Give an explicit formula for Δx.

(j)

How does this primal–dual affine-scaling algorithm compare with the iteratively reweighted least squares algorithm defined in Section 12.5?

18.9

(a)

Let ξ j , $$j = 1,2,\ldots$$, denote a sequence of real numbers between zero and one. Show that $$\prod _{j}(1 -\xi _{j}) = 0$$ if $$\sum _{j}\xi _{j} = \infty $$.

(b)

Use the result of part 18.9a to prove the following convergence result: if the sequences $$\|{x}^{(k)}\|_{\infty }$$, $$k = 1,2,\ldots$$, and $$\|{y}^{(k)}\|_{\infty }$$, $$k = 1,2,\ldots$$, are bounded and $$\sum {_{k}\theta }^{(k)} = \infty $$, then

Notes

The path-following algorithm introduced in this chapter has its origins in a paper by Kojima et al. (1989). Their paper assumed an initial feasible solution and therefore was a true interior-point method. The method given in this chapter does not assume the initial solution is feasible—it is a one-phase algorithm. The simple yet beautiful idea of modifying the Kojima–Mizuno–Yoshise primal–dual algorithm to make it into a one-phase algorithm is due to Lustig (1990).

Of the thousands of papers on interior-point methods that have appeared in the last decade, the majority have included convergence proofs for some version of an interior-point method. Here, we only mention a few of the important papers. The first polynomial-time algorithm for linear programming was discovered by Khachian (1979). Khachian’s algorithm is fundamentally different from any algorithm presented in this book. Paradoxically, it proved in practice to be inferior to the simplex method. N.K. Karmarkar’s pathbreaking paper (Karmarkar 1984) contained a detailed convergence analysis. His claims, based on preliminary testing, that his algorithm is uniformly substantially faster than the simplex method sparked a revolution in linear programming. Unfortunately, his claims proved to be exaggerated, but nonetheless interior-point methods have been shown to be competitive with the simplex method and usually superior on very large problems. The convergence proof for a primal–dual interior-point method was given by Kojima et al. (1989). Shortly thereafter, Monteiro and Adler (1989) improved on the convergence analysis. Two recent survey papers, Todd (1995) and Anstreicher (1996), give nice overviews of the current state of the art. Also, a soon-to-be-published book by Wright (1996) should prove to be a valuable reference to the reader wishing more information on convergence properties of these algorithms.

The homotopy method outlined in Exercise 18.5 is described in Nazareth (1986, 1996). Higher-order path-following methods are described (differently) in Carpenter et al. (1993).

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Footnotes

1

For compactness, we use the notation a ∧ b to represent the minimum of the two numbers a and b.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_19

19. The KKT System

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

The most time-consuming aspect of each iteration of the path-following method is solving the system of equations that defines the step direction vectors Δx, Δy, Δw, and Δz:

(19.1)

(19.2)

(19.3)

(19.4)

After minor manipulation, these equations can be written in block matrix form as follows:

(19.5)

This system is called the Karush–Kuhn–Tucker system, or simply the KKT system. It is a symmetric linear system of 2n + 2m equations in 2n + 2m unknowns. One could, of course, perform a factorization of this large system and then follow that with a forward and backward substitution to solve the system. However, it is better to do part of this calculation “by hand” first and only use a factorization routine to help solve a smaller system. There are two stages of reductions that one could apply. After the first stage, the remaining system is called the reduced KKT system, and after the second stage it is called the system of normal equations. We shall discuss these two systems in the next two sections.

1 The Reduced KKT System

Equations (19.3) and (19.4) are trivial (in the sense that they only involve diagonal matrices), and so it seems sensible to eliminate them right from the start. To preserve the symmetry that we saw in (19.5), we should solve them for Δz and Δw, respectively:

Substituting these formulas into (19.1) and (19.2), we get the so-called reduced KKT system:

(19.6)

(19.7)

Substituting in the definitions of ρ and σ and writing the system in matrix notation, we get

Note that the reduced KKT matrix is again a symmetric matrix. Also, the right-hand side displays symmetry between the primal and the dual. To reduce the system any further, one needs to break the symmetry that we have carefully preserved up to this point. Nonetheless, we forge ahead.

2 The Normal Equations

For the second stage of reduction, there are two choices: we could either (1) solve (19.6) for Δy and eliminate it from (19.7) or (2) solve (19.7) for Δx and eliminate it from (19.6). For the moment, let us assume that we follow the latter approach. In this case, we get from (19.7) that

(19.8)

which we use to eliminate Δx from (19.6) to get

(19.9)

This last system is a system of m equations in m unknowns. It is called the system of normal equations in primal form. It is a system of equations involving the matrix $${Y }^{-1}W + AX{Z}^{-1}{A}^{T}$$. The Y  − 1 W term is simply a diagonal matrix, and so the real meat of this matrix is contained in the AXZ  − 1 A T term.

Given that A is sparse (which is generally the case in real-world linear programs), one would expect the matrix AXZ  − 1 A T to be likewise sparse. However, we need to investigate the sparsity of AXZ  − 1 A T (or lack thereof) more closely. Note that the (i, j)th element of this matrix is given by

That is, the (i, j)th element is simply a weighted inner product of rows i and j of the A matrix. If these rows have disjoint nonzero patterns, then this inner product is guaranteed to be zero, but otherwise it must be treated as a potential nonzero. This is bad news if A is generally sparse but has, say, one dense column:

But don’t forget that we didn’t have to go the primal normal equations route. Instead, we could have chosen the other alternative of solving (19.6) for Δy,

and eliminating it from (19.7):

(19.10)

The system defined by (19.10) is a system of n equations in n unknowns. It is called the system of normal equations in dual form. Note that dense columns do not pose a problem for these equations. Indeed, for the example given above, we now get

While this system is larger than the one before, it is also sparse, and sparsity almost always is more important than matrix dimensions. In this example, the dense matrix associated with the primal normal equations requires 65 arithmetic operations to factor, whereas the larger, sparser matrix associated with the dual normal equations requires just 60. This is a small difference, but these are small matrices. As the matrices involved get large, factoring a dense matrix takes on the order of n ³ operations, whereas a very sparse matrix might take only on the order of n operations. Clearly, as n gets large, the difference between these becomes quite significant.

It would be great if we could say that it is always best to solve the primal normal equations or the dual normal equations. But as we’ve just seen, dense columns in A are bad for the primal normal equations and, of course, it follows that dense rows are bad for the dual normal equations. Even worse, some problems have constraint matrices A that are overall very sparse but contain some dense rows and some dense columns. Such problems are sure to run into trouble with either sets of normal equations. For these reasons, it is best to factor the matrix in the reduced KKT system directly. Then it is possible to find pivot orders that circumvent the difficulties posed by both dense columns and dense rows.

3 Step Direction Decomposition

In the next chapter, we shall discuss factorization techniques for symmetric matrices (along with other implementation issues). However, before we embark on that discussion, we end this chapter by taking a closer look at the formulas for the step direction vectors. To be specific, let us look at Δx. From the primal normal equations (19.9), we can solve for Δy and then substitute the solution into (19.8) to get an explicit formula for Δx:

(19.11)

where we have denoted by D the positive diagonal matrix defined by

and we have denoted by E the positive diagonal matrix defined by

(defining these matrices by their squares is possible, since the squares have positive diagonal entries). However, using the dual normal equations, we get

(19.12)

These two expressions for Δx look entirely different, but they must be the same, since we know that Δx is uniquely defined by the reduced KKT system. They are indeed the same, as can be shown directly by establishing a certain matrix identity. This is the subject of Exercise 19.1. There are a surprising number of published research papers on interior-point methods that present supposedly new algorithms that are in fact identical to existing methods. These papers get published because the equivalence is not immediately obvious (such as the one we just established).

We can gain further insight into the path-following method by looking more closely at the primal step direction vector. Formula (19.11) can be rearranged as follows:

For comparison purposes down the road, we prefer to write the Y  − 1 e that appears in the second term containing μ as $${E}^{-2}{W}^{-1}e$$. Also, using the result of Exercise 19.2, we can rewrite the bulk of the last line as follows:

Putting this all together, we get

where

and

In Chapter 21, we shall show that these components of Δx have important connections to the step directions that arise in a related interior-point method called the affine-scaling method. For now, we simply establish some of their properties as they relate to the path-following method. Our first result is that Δx opt is an ascent direction.

Theorem 19.1.

c T Δx opt ≥ 0.

Proof.

We use the result of Exercise 19.1 (with the roles of E and D switched) to see that

Hence,

We claim that the right-hand side is obviously nonnegative, since the matrix sandwiched between c and its transpose is positive semidefinite.¹ Indeed, the claim follows from the definition of positive semidefiniteness: a matrix B is positive semidefinite if ξ T Bξ ≥ 0 for all vectors ξ. To see that the matrix in question is in fact positive semidefinite, we first note that A T E ² A and D  − 2 are positive semidefinite:

Then we show that the sum of two positive semidefinite matrices is positive semidefinite and finally that the inverse of a symmetric positive semidefinite matrix is positive semidefinite. To verify closure under summation, suppose that B ⁽¹⁾ and B ⁽²⁾ are positive semidefinite, and then compute

To verify closure under forming inverses of symmetric positive semidefinite matrices, suppose that B is symmetric and positive semidefinite. Then

where the inequality follows from the fact that B is positive semidefinite and B  − 1 ξ is simply any old vector. This completes the proof. □ 

The theorem just proved justifies our referring to Δx opt as a step-toward-optimality direction. We next show that Δx feas is in fact a step-toward-feasibility.

In Exercise 19.3, you are asked to find the formulas for the primal slack vector’s step directions, Δw opt, Δw ctr, and Δw feas. It is easy to verify from these formulas that the pairs (Δx opt, Δw opt) and (Δx ctr, Δw ctr) preserve the current level of infeasibility. That is,

and

Hence, only the “feasibility” directions can improve the degree of feasibility. Indeed, it is easy to check that

Finally, we consider Δx ctr. If the objective function were zero (i.e., c = 0) and if the current point were feasible, then steps toward optimality and feasibility would vanish and we would be left with just Δx ctr. Since our step directions were derived in an effort to move toward a point on the central path parametrized by μ, we now see that Δx ctr plays the role of a step-toward-centrality.

Exercises

19.1

Sherman–Morrison–Woodbury Formula. Assuming that all the inverses below exist, show that the following identity is true:

Use this identity to verify directly the equivalence of the expressions given for Δx in (19.11) and (19.12).

19.2

Assuming that all the inverses exist, show that the following identity holds:

19.3

Show that

where

and

Notes

The KKT system for general inequality constrained optimization problems was derived by Kuhn and Tucker (1951). It was later discovered that W. Karush had proven the same result in his 1939 master’s thesis at the University of Chicago (Karush 1939). John (1948) was also an early contributor to inequality-constrained optimization. Kuhn’s survey paper (Kuhn 1976) gives a historical account of the development of the subject.

Bibliography

John, F. (1948). Extremum problems with inequalities as subsidiary conditions. In K. Fredrichs, O. Neugebauer, and J. Stoker (Eds.), Studies and essays: Courant anniversary volume (pp. 187–204). New York: Wiley.

Karush, W. (1939). Minima of functions of several variables with inequalities as side conditions (Technical report, M.S. thesis) Department of Mathematics, University of Chicago.

Kuhn, H. (1976). Nonlinear prgramming: A historical view. In R. Cottle and C. Lemke (Eds.), Nonlinear programming, SIAM-AMS proceedings (Vol. 9, pp. 1–26). Providence: American Mathetical Society.

Kuhn, H., and Tucker, A. (1951). Nonlinear prgramming. In J. Neyman (Ed.), Proceedings of the second Berkeley symposium on mathematical statistics and probability (pp. 481–492). Berkeley: University of California Press.

Footnotes

1

In fact, it’s positive definite, but we don’t need this stronger property here.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_20

20. Implementation Issues for Interior-Point Methods

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In this chapter, we discuss implementation issues that arise in connection with the path-following method.

The most important issue is the efficient solution of the systems of equations discussed in the previous chapter. As we saw, there are basically three choices, involving either the reduced KKT matrix,

(20.1)

or one of the two matrices associated with the normal equations:

(20.2)

or

(20.3)

(Here, $${E}^{-2} = {Y }^{-1}W$$ and $${D}^{-2} = {X}^{-1}Z$$.)

In the previous chapter, we explained that dense columns/rows are bad for the normal equations and that therefore one might be better off solving the system involving the reduced KKT matrix. But there is also a reason one might prefer to work with one of the systems of normal equations. The reason is that these matrices are positive definite. We shall show in the first section that there are important advantages in working with positive definite matrices. In the second section, we shall consider the reduced KKT matrix and see to what extent the nice properties possessed by positive definite matrices carry over to these matrices.

After finishing our investigations into numerical factorization, we shall take up a few other relevant tasks, such as how one extends the path-following algorithm to handle problems with bounds and ranges.

1 Factoring Positive Definite Matrices

As we saw in the proof of Theorem 19.1, the matrix (20.2) appearing in the primal normal equations is positive semidefinite (and so is (20.3), of course). In fact, it is even better—it’s positive definite. A matrix B is positive definite if ξ T Bξ > 0 for all vectors ξ≠0. In this section, we will show that, if we restrict our row/column reordering to symmetric reorderings, that is, reorderings where the rows and columns undergo the same permutation, then there is no danger of encountering a pivot element whose value is zero. Hence, the row/column permutation can be selected ahead of time based only on the aim of maintaining sparsity.

If we restrict ourselves to symmetric permutations, each pivot element is a diagonal element of the matrix. The following result shows that we can start by picking an arbitrary diagonal element as the first pivot element:

Theorem 20.1.

If B is positive definite, then b ii > 0 for all i.

The proof follows trivially from the definition of positive definiteness:

The next step is to show that after each stage of the elimination process, the remaining uneliminated matrix is positive definite. Let us illustrate by breaking out the first row/column of the matrix and looking at what the first step of the elimination process does. Breaking out the first row/column, we write

Here, a is the first diagonal element (a scalar), b is the column below a, and C is the matrix consisting of all of B except the first row/column. One step of elimination (as described in Chapter 8) transforms B into

The following theorem tells us that the uneliminated part is positive definite:

Theorem 20.2.

If B is positive definite, then so is $$C - b{b}^{T}/a$$ .

Proof.

The fact that B is positive definite implies that

(20.4)

is positive whenever the scalar x or the vector y is nonzero (or both). Fix a vector y≠0, and put $$x = -\frac{1} {a}{b}^{T}y$$. Using these choices in (20.4), we get

Since y was an arbitrary nonzero vector, it follows that $$C - b{b}^{T}/a$$ is positive definite. □ 

Hence, after one step of the elimination, the uneliminated part is positive definite. It follows by induction then that the uneliminated part is positive definite at every step of the elimination.

Here’s an example:

At the end of the four steps of the elimination (without permutations), we end up with

From this eliminated form, we extract the lower triangular matrix, the diagonal matrix, and the upper triangular matrix to write B as

As we saw in Chapter 8, it is convenient to combine the lower triangular matrix with the diagonal matrix to get a new lower triangular matrix with ones on the diagonal. But the current lower triangular matrix is exactly the transpose of the upper triangular matrix. Hence, to preserve symmetry, we should combine the diagonal matrix with both the lower and the upper triangular matrices. Since it only appears once, we must multiply and divide by it (in the middle of the product). Doing this, we get

The lower triangular matrix in this representation is usually denoted by L and the diagonal matrix by D (not to be confused with the D at the beginning of the chapter). Hence, this factorization can be summarized as

and is referred to as an LDL T -factorization. Of course, once a factorization is found, it is easy to solve systems of equations using forward and backward substitution as discussed in Chapter 8.

1.1 Stability

We began our discussion of factoring positive definite matrices with the comment that a symmetric permutation can be chosen purely with the aim of preserving sparsity, since it is guaranteed that no pivot element will ever vanish. However, the situation is even better than that—we can show that whenever a pivot element is small, so is every other nonzero in the uneliminated part of the same row/column. Before saying why, we need to set down a few technical results.

Theorem 20.3.

If $$\bar{b}_{ii}$$ denotes a diagonal element in the uneliminated submatrix at some stage of an elimination and b ii denotes the original value of that diagonal element, then $$0 <\bar{ b}_{ii} \leq b_{ii}$$.

Proof.

The positivity of $$\bar{b}_{ii}$$ follows from the fact the uneliminated submatrix is positive definite. The fact that it is bounded above by b ii follows from the fact that each step of the elimination can only decrease diagonal elements, which can be seen by looking at the first step of the elimination. Using the notation introduced just after Theorem 20.1,

 □ 

Theorem 20.4.

If B is symmetric and positive definite, then $$\vert b_{ij}\vert < \sqrt{b_{ii } b_{jj}}$$ for all i≠j.

Proof.

Fix i≠j and let $$\xi = re_{i} + e_{j}$$. That is, ξ is the vector that’s all zero except for the ith and jth position, where it’s r and 1, respectively. Then,

for all $$r \in \mathbb{R}$$. This quadratic expression is positive for all values of r if and only if it is positive at its minimum, and it’s easy to check that it is positive at that point if and only if $$\vert b_{ij}\vert < \sqrt{b_{ii } b_{jj}}$$. □ 

These two theorems, together with the fact that the uneliminated submatrix is symmetric and positive definite, give us bounds on the off-diagonal elements. Indeed, consider the situation after a number of steps of the elimination. Using bars to denote matrix elements in the uneliminated submatrix and letting M denote an upper bound on the diagonal elements before the elimination process began (which, without loss of generality, could be taken as 1), we see that, if $$\bar{b}_{jj} <\epsilon$$, then

(20.5)

This bound is exceedingly important and is special to positive definite matrices.

2 Quasidefinite Matrices

In this section, we shall study factorization techniques for the reduced KKT matrix (20.1). The reduced KKT matrix is an example of a quasidefinite matrix. A symmetric matrix is called quasidefinite if it can be written (perhaps after a symmetric permutation) as

where E and D are positive definite matrices. Quasidefinite matrices inherit some of the nice properties of positive definite matrices. In particular, one can perform an arbitrary symmetric permutation of the rows/columns and still be able to form a factorization of the permuted matrix.

The idea is that, after each step of the elimination, the remaining uneliminated part of the matrix is still quasidefinite. To see why, let’s break out the first row/column of the matrix and look at the first step of the elimination process. Breaking out the first row/column of B, we write

where a is a scalar, b and f are vectors, and C, D, and G are matrices (of the appropriate dimensions). One step of the elimination process transforms B into

The uneliminated part is

Clearly, the lower-left and upper-right blocks are transposes of each other. Also, the upper-left and lower-right blocks are symmetric, since C and D are. Therefore, the whole matrix is symmetric. Theorem 20.2 tells us that $$C - b{b}^{T}/a$$ is positive definite and $$D + f{f}^{T}/a$$ is positive definite, since the sum of a positive definite matrix and a positive semidefinite matrix is positive definite (see Exercise 20.2). Therefore, the uneliminated part is indeed quasidefinite.

Of course, had the first pivot element been selected from the submatrix D instead of E, perhaps the story would be different. But it is easy to check that it’s the same. Hence, no matter which diagonal element is selected to be the first pivot element, the resulting uneliminated part is quasidefinite. Now, by induction it follows that every step of the elimination process involves choosing a pivot element from the diagonals of a quasidefinite matrix. Since these diagonals come from either a positive definite submatrix or the negative of such a matrix, it follows that they are always nonzero (but many of them will be negative). Therefore, just as for positive definite matrices, an arbitrary symmetric permutation of a quasidefinite matrix can be factored without any risk of encountering a zero pivot element.

Here’s an example:

(20.6)

(The blocks are easy to pick out, since the negative diagonals must be from − E, whereas the positive ones are from D.) Let’s eliminate by picking the diagonals in the order 1, 5, 2, 4, 3. No permutations are needed in preparation for the first step of the elimination. After this step, we have

Now, we move row/column 5 to the pivot position, slide the other rows/columns down/over, and eliminate to get

Row/column 2 is in the correct position for the third step of the elimination, and therefore, without further ado, we do the next step in the elimination:

Finally, we interchange rows/columns 3 and 4 and do the last elimination step to get

From this final matrix, it is easy to extract the LDL T -factorization of the permutation of B:

As always, the factorization is computed so that one can solve systems of equations. Given the factorization, all that is required is a forward substitution, a backward substitution, a scaling, and two permutations.

2.1 Instability

We have shown that an arbitrary symmetric permutation of the rows/columns of a quasidefinite matrix can be factored into LDL T . That is, mathematically none of the pivot elements will be zero. But they can be small, and their smallness can cause troubles not encountered with positive definite matrices. To explain, let’s look at an example. Consider the linear programming problem

and its dual

Drawing pictures, it is easy to see that the optimal solution is

Therefore, as the optimal solution is approached, the diagonal elements in X  − 1 Z and Y  − 1 W approach zero. Therefore, at late stages in the path-following method, one is faced with the problem of factoring the following reduced KKT matrix:

where ε ₁, ε ₂, δ ₁, and δ ₂ are small positive numbers. Consider what happens if we permute the rows/columns so that the original diagonal elements appear in the following order: 1, 3, 4, 2. The permuted matrix is seen to be

After the first two steps of the elimination, we have

(20.7)

Using exact arithmetic, the uneliminated part simplifies to

Here, the diagonal elements are small but not zero and the off-diagonal elements are close to − 3. But on a computer that stores numbers with finite precision, the computation comes out quite differently when the ε i ’s and the δ i ’s are smaller than the square root of the machine precision (i.e., about 10 − 8 for double-precision floating-point numbers). In the elimination process, the parenthesized expressions in (20.7) are evaluated before the other operations. Hence, in finite precision arithmetic, these expressions produce

and so (20.7) becomes

which clearly presents a problem for the next step in the elimination process.

Now let’s consider what happens if the elimination process is applied directly to B without permuting the rows/columns. Sparing the reader the tedious details, the end result of the elimination is the following matrix:

(20.8)

As before, in finite precision arithmetic, certain small numbers get lost:

Making these substitutions in (20.8), we see that the final matrix produced by the elimination process using finite precision arithmetic is

Just as before, the fact that small numbers got lost has resulted in a zero appearing on the diagonal where a small but nonzero (in this case positive) number belongs. However, the situation is fundamentally different this time. With the first ordering, a − 3 remained to be eliminated under the zero diagonal element, whereas with the second ordering, this did not happen. Of course, it didn’t happen in this particular example because the 0 appeared as the last pivot element, which has no elements below it to be eliminated. But that is not the general reason why the second ordering does not produce nonzeros under zero pivot elements. In general, a zero (which should be a small positive) can appear anywhere in the lower-right block (relative to the original quasidefinite partitioning). But once the elimination process gets to this block, the remaining uneliminated part of the matrix is positive definite. Hence, the estimate in (20.5) can be used to tell us that all the nonzeros below a zero diagonal are in fact small themselves. A zero appearing on the diagonal only presents a problem if there are nonzeros below it that need to be eliminated. If there are none, then the elimination can simply proceed to the next pivot element (see Exercise 20.1).

Let’s summarize the situation. We showed in the last chapter that the possibility of dense rows/columns makes it unattractive to work strictly with the normal equations. Yet, although the quasidefinite reduced KKT system can be used, it is numerically less stable. A compromise solution seems to be suggested. One could take a structured approach to the reordering heuristic. In the structured approach, one decides first whether it seems better to begin pivoting with elements from the upper-left block or from the lower-right block. Once this decision is made, one should pivot out all the diagonal elements from this block before working on the other block, with the exception that pivots involving dense rows/columns be deferred to the end of the elimination process. If no dense columns are identified, this strategy mimics the normal equations approach. Indeed, after eliminating all the diagonal elements in the upper-left block, the remaining uneliminated lower-right block contains exactly the matrix for the system of dual normal equations. Similarly, had the initial choice been to pivot out all the diagonal elements from the lower-right block, then the remaining uneliminated upper-left block becomes the matrix for the system of primal normal equations.

With this structured approach, if no dense rows/columns are identified and deferred, then the elimination process is numerically stable. If, on the other hand, some dense rows/columns are deferred, then the factorization is less stable. But in practice, this approach seems to work well. Of course, one could be more careful and monitor the diagonal elements. If a diagonal element gets small (relative to the other uneliminated nonzeros in the same row/column), then one could flag it and then calculate a new ordering in which such pivot elements are deferred to the end of the elimination process.

3 Problems in General Form

In this section, we describe how to adapt the path-following algorithm to solving problems presented in the following general form:

(20.9)

As in Chapter 9, some of the data elements are allowed to take on infinite values. However, let us consider first the case where all the components of a, b, l, and u are finite. Infinities require special treatment, which shall be discussed shortly.

Following the derivation of the path-following method that we introduced in Chapter 18, the first step is to introduce slack variables as appropriate to replace all inequality constraints with simple nonnegativity constraints. Hence, we rewrite the primal problem (20.9) as follows:

In Chapter 9, we showed that the dual problem is given by

and the corresponding complementarity conditions are given by

The next step in the derivation is to introduce the primal–dual central path, which we parametrize as usual by a positive real parameter μ. Indeed, for each μ > 0, we define the associated central-path point in primal–dual space as the unique point that simultaneously satisfies the conditions of primal feasibility, dual feasibility, and μ-complementarity. Ignoring nonnegativity (which is enforced separately), these conditions are

Note that we have replaced the primal feasibility condition, $$-Ax + p = -a$$, with the equivalent condition that $$f + p = b - a$$, and we have introduced into the dual problem new variables y defined by $$y = v - q$$. The reason for these changes is to put the system of equations into a form in which A and A T appear as little as possible (so that solving the system of equations for the step direction will be as efficient as possible).

The last four equations are the μ-complementarity conditions. As usual, each upper case letter that appears on the left in these equations denotes the diagonal matrix having the components of the corresponding lower-case vector on its diagonal. The system is a nonlinear system of 5n + 5m equations in 5n + 5m unknowns. It has a unique solution in the strict interior of the following subset of primal–dual space:

(20.10)

This fact can be seen by noting that these equations are the first-order optimality conditions for an associated strictly convex barrier problem.

As μ tends to zero, the central path converges to the optimal solution to both the primal and dual problems. The path-following algorithm is defined as an iterative process that starts from a point in the strict interior of (20.10), estimates at each iteration a value of μ representing a point on the central path that is in some sense closer to the optimal solution than the current point, and then attempts to step toward this central-path point, making sure that the new point remains in the strict interior of the set given in (20.10).

Suppose for the moment that we have already decided on the target value for μ. Let (x, …, h) denote the current point in the strict interior of (20.10), and let $$(x + \Delta x,\ldots ,h + \Delta h)$$ denote the point on the central path corresponding to the target value of μ. The defining equations for the point on the central path can be written as

where we have introduced notations ρ, …, γ h as shorthands for the right-hand side expressions. This is almost a linear system for the direction vectors (Δx, …, Δh). The only nonlinearities appear on the right-hand sides of the complementarity equations (i.e., in γ f , …, γ h ). As we saw before, Newton’s method suggests that we simply drop these nonlinear terms to obtain a linear system for the “delta” variables.

Clearly, the main computational burden is to solve the system shown above. It is important to note that this is a large, sparse, indefinite, linear system. It is also symmetric if one negates certain rows and rearranges rows and columns appropriately:

Because this system is symmetric, we look for symmetric pivots to solve it. That is, we choose our pivots only from the diagonal elements. It turns out that we can eliminate Δv, Δp, Δg, and Δt using the nonzero diagonals $$-F{V }^{-1}$$, QP  − 1, HG  − 1, and ST  − 1, respectively, in any order without causing any nondiagonal fill-in. Indeed, the equations for Δv, Δp, Δg, and Δt are

(20.11)

and after elimination from the system, we get

where again we have introduced abbreviated notations for the components of the right-hand side:

Next we use the pivot elements $$-T{S}^{-1}$$, $$-G{H}^{-1}$$, and $$-P{Q}^{-1}$$ to solve for Δs, Δh, and Δq, respectively:

(20.12)

After eliminating these variables, the system simplifies to

where

and

Finally, we use the pivot element E to solve for Δf,

(20.13)

which brings us to the reduced KKT equations:

(20.14)

Up to this point, none of the eliminations have produced any off-diagonal fill-in. Also, the matrix for system given in (20.14) is a symmetric quasidefinite matrix. Hence, the techniques given in Section 19.2 for solving such systems can be used. The algorithm is summarized in Figure 20.1.

Fig. 20.1

The path-following method—general form.

Exercises

20.1

The matrix

is not positive definite but is positive semidefinite. Find a factorization B = LDL T , where L is lower triangular with ones on the diagonal and D is a diagonal matrix with nonnegative diagonal elements. If such a factorization exists for every symmetric positive semidefinite matrix, explain why. If not, give a counterexample.

20.2

Show that the sum of a positive definite matrix and a positive semidefinite matrix is positive definite.

20.3

Permute the rows/columns of the matrix B given in (20.6) so that the diagonal elements from B appear in the order 2, 3, 4, 5, 1. Compute an LDL T -factorization of this matrix.

20.4

Show that, if B is symmetric and positive semidefinite, then $$\vert b_{ij}\vert \leq \sqrt{b_{ii } b_{jj}}$$ for all i, j.

Notes

Most implementations of interior-point methods assume the problem to be formulated with equality constraints. In this formulation, Lustig et al. (1994) give a good overview of the performance of interior-point algorithms compared with the simplex method.

The suggestion that it is better to solve equations in the KKT form instead of normal form was offered independently by a number of researchers (Gill et al. 1992; Turner 1991; Fourer and Mehrotra 1991; Vanderbei and Carpenter 1993).

The advantages of the primal–dual symmetric formulation were first reported in Vanderbei (1994). The basic properties of quasidefinite matrices were first given in Vanderbei (1995).

References

Fourer, R., and Mehrotra, S. (1991). Solving symmetric indefinite systems in an interior point method for linear programming. Mathematical Programming, 62, 15–40.Crossref

Gill, P., Murray, W., Ponceleón, D., and Saunders, M. (1992). Preconditioners for indefinite systems arising in optimization. SIAM Journal on Matrix Analysis and Applications, 13(1), 292–311.Crossref

Lustig, I., Marsten, R., and Shanno, D. (1994). Interior point methods for linear programming: Computational state of the art. ORSA Journal on Computing, 6, 1–14.Crossref

Turner, K. (1991). Computing projections for the Karmarkar algorithm. Linear Algebra and Its Applications, 152, 141–154.Crossref

Vanderbei, R. (1994). Interior-point methods: Algorithms and formulations. ORSA Journal on Computing, 6, 32–34.Crossref

Vanderbei, R. (1995). Symmetric quasi-definite matrices. SIAM Journal on Optimization, 5(1), 100–113.Crossref

Vanderbei, R., and Carpenter, T. (1993). Symmetric indefinite systems for interior-point methods. Mathematical Programming, 58, 1–32.Crossref

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_21

21. The Affine-Scaling Method

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In the previous chapter, we showed that the step direction for the path-following method can be decomposed into a linear combination of three directions: a direction toward optimality, a direction toward feasibility, and a direction toward centrality. It turns out that these directions, or minor variants of them, arise in all interior-point methods.

Historically, one of the first interior-point methods to be invented, analyzed, and implemented was a two-phase method in which Phase I uses only the feasibility direction and Phase II uses only the optimality direction. This method is called affine scaling. While it is no longer considered the method of choice for practical implementations, it remains important because its derivation provides valuable insight into the nature of the three basic directions mentioned above.

In this chapter, we shall explain the affine-scaling principle and use it to derive the step toward optimality and step toward feasibility directions. As always, our main interest lies in problems presented in standard form. But for affine scaling, it is easier to start by considering problems in equality form. Hence, we begin by assuming that the linear programming problem is given as

(21.1)

We shall begin with the Phase II algorithm. Hence, we assume that we have a feasible initial starting point, x ⁰. For the affine-scaling method, it is important that this starting point lie in the strict interior of the feasible set. That is, we assume that

1 The Steepest Ascent Direction

Since the affine-scaling principle is fundamentally geometric, it is useful to keep a picture in mind. A typical picture for m = 1 and n = 3 is shown in Figure 21.1. The ultimate goal is, of course, to move from x ⁰ to the optimal solution x  ∗ . However, the short-term goal is to move from x ⁰ in some direction Δx that improves the objective function. Such a direction is called an ascent direction. You probably recall from elementary calculus that the best, i.e., steepest, ascent direction is given by the gradient of the objective function. However, as we see in Figure 21.1, there is no reason a priori for the gradient to “lie in” the feasible region. Hence, the steepest ascent direction will almost surely cause a move to infeasible points. This is also clear algebraically. Indeed,

(unless Ac = 0 which is not likely).

FIGURE 21.1

A typical feasible region when the problem is in equality form, m = 1, and n = 3. The lines drawn on the feasible set represent level sets of the objective function, and x ⁰ represents the starting point for the affine-scaling method.

To see how to find a better direction, let us first review in what sense the gradient is the steepest ascent direction. The steepest ascent direction is defined to be the direction that gives the greatest increase in the objective function subject to the constraint that the displacement vector has unit length. That is, the steepest ascent direction is the solution to the following optimization problem:

(21.2)

We can solve this problem using Lagrange multipliers. Indeed, if we let λ denote the Lagrange multiplier for the constraint, the problem becomes

Differentiating with respect to Δx and setting the derivative to zero, we get

which implies that

Then differentiating the Lagrangian with respect to λ and setting that derivative to zero, we see that

which implies that

Hence, the steepest ascent direction points in the direction of either c or its negative. Since the negative is easily seen not to be an ascent direction at all, it follows that the steepest ascent direction points in the direction of c.

2 The Projected Gradient Direction

The problem with the steepest ascent direction is that it fails to preserve feasibility. That is, it fails to preserve the equality constraints Ax = b. To remedy this problem, let’s add these constraints to (21.2) so that we get the following optimization problem:

Again, the method of Lagrange multipliers is the appropriate tool. As before, let λ denote the Lagrange multiplier for the norm constraint, and now introduce a vector y containing the Lagrange multipliers for the equality constraints. The resulting unconstrained optimization problem is

Differentiating this Lagrangian with respect to Δx, λ, and y and setting these derivatives to zero, we get

The second equation tells us that the length of Δx is one. Since we are interested in the direction of Δx and are not concerned about its length, we ignore this second equation. The first equation tells us that Δx is proportional to c − A T y, and again, since we aren’t concerned about lengths, we put $$\lambda = 1/2$$ so that the first equation reduces to

(21.3)

Since Ax ⁰ = b, the third equation says that

Substituting (21.3) into this equation, we get

which, assuming that AA T has full rank (as it should), can be solved for y to get

Now, substituting this expression into (21.3), we see that

It is convenient to let P be the matrix defined by

With this definition, Δx can be expressed succinctly as

We claim that P is the matrix that maps any vector, such as c, to its orthogonal projection onto the null space of A. To justify this claim, we first need to define some of the terms we’ve used. The null space of A is defined as {d ∈ ℝ n : Ad = 0}. We shall denote the null space of A by N(A). A vector $$\tilde{c}$$ is the orthogonal projection of c onto N(A) if it lies in the null space,

and if the difference between it and c is orthogonal to every other vector in N(A). That is,

Hence, to show that Pc is the orthogonal projection of c onto the null space of A, we simply check these two conditions. Checking the first, we see that

which clearly vanishes. To check the second condition, let d be an arbitrary vector in the null space, and compute

which also vanishes, since $${d}^{T}{A}^{T} = {(Ad)}^{T} = 0$$. The orthogonal projection Pc is shown in Figure 21.1.

3 The Projected Gradient Direction with Scaling

The orthogonal projection of the gradient gives a good step direction in the sense that among all feasibility-preserving directions, it gives the largest rate of increase of c T x per unit step length. This property is nice as long as the current point x ⁰ is well inside the feasible set. But if it is close to a “wall,” the overall increase in one step will be small, since even though the rate is large the actual step length will be small, yielding a small overall increase. In fact, the increase will become arbitrarily small as the point x ⁰ is taken closer and closer to a “wall.” Hence, to get a reasonable algorithm, we need to find a formula for computing step directions that steers away from walls as they get close.

The affine-scaling algorithm achieves this affect as follows: scale the variables in the problem so that the current feasible solution is far from the walls, compute the step direction as the projected gradient in the scaled problem, and then translate this direction back into the original system. The idea of scaling seems too simple to do any good, and this is true if one tries the most naive scaling—just multiplying every variable by one large number (such as the reciprocal of the smallest component of x ⁰). Such a uniform scaling does not change the picture in any way. For example, Figure 21.1, which doesn’t show specific scales on the coordinate axes, would not change at all. Hence, whether distance is measured in miles or in feet, the property of being close to a wall remains unchanged.

Fortunately, the scaling we have in mind for the affine-scaling algorithm is just slightly fancier. Indeed, the idea is to scale each variable in such a manner that its initial value gets mapped to 1. That is, for each j = 1, 2, …, n, we introduce new variables given by

Of course, this change of variable is trivial to undo:

In matrix notation, the change of variables can be written as

(21.4)

Note that we are employing our usual convention of letting an upper-case letter stand for a diagonal matrix whose diagonal elements come from the components of the vector denoted by the corresponding lower-case letter. Clearly, under this change of variables, the initial solution x ⁰ gets mapped to the vector e of all ones, which is at least one unit away from each wall. Figure 21.2 shows an example of this scaling transformation. Note that, unlike the trivial scaling mentioned above, this scaling changes the way the level sets of the objective cut across the feasible region. Making the change of variables given by (21.4) in (21.1), we find that the problem in the scaled space is

Clearly, it is a linear programming problem in standard form with constraint matrix AX ⁰ and vector of objective function coefficients $${({c}^{T}{X}^{0})}^{T} = {X}^{0}c$$. Letting Δξ denote the projected gradient of the objective function in this scaled problem, we see that

Ignore step length worries for the moment, and consider moving from the current solution ξ ⁰ = e to the following new point in the scaled problem:

Then transforming this new point back into the original unscaled variables, we get a new point x ¹ given by

Of course, the difference between x ¹ and x ⁰ is the step direction in the original variables. Denoting this difference by Δx, we see that

(21.5)

where

The expression for Δx given by (21.5) is called the affine-scaling step direction. Of course, to construct the affine-scaling algorithm out of this formula for a step direction, one simply needs to choose step lengths in such a manner as to ensure “strict” feasibility of each iterate.

FIGURE 21.2

The effect of affine scaling on projected gradients.

We end this section by illustrating some of the calculations on the following trivial example:

This is precisely the problem shown in Figure 21.2. As in the figure, let us assume that

For later comparison, we compute the projected gradient (without scaling):

Now, in the scaled coordinate system, the gradient of the objective function is

and the constraint matrix is given by

Using these, we compute Δξ as follows:

Finally, Δx is obtained by the inverse scaling:

In the next section, we discuss the convergence properties of the affine-scaling algorithm.

4 Convergence

In the previous section, we derived the affine-scaling step direction Δx. To make an algorithm, we need to introduce an associated step length. If the step were chosen so that the new point were to lie exactly on the boundary of the feasible region, then the multiplier for Δx, which as usual we denote by θ, would be given by

But as always, we need to shorten the step by introducing a parameter 0 < r < 1 and setting

With this choice of θ, the iterations of the affine-scaling algorithm are defined by

It turns out that the analysis of the affine-scaling algorithm is more delicate than the analysis of the path-following algorithm that we discussed in Chapter 18. Hence, we simply state without proof the main results.

Theorem 21.1.

(a)

If the problem and its dual are nondegenerate, then for every r < 1, the sequence generated by the algorithm converges to the optimal solution.

(b)

For r ≤ 2∕3, the sequence generated by the algorithm converges to an optimal solution (regardless of degeneracy).

(c)

There exists an example and an associated r < 1 for which the algorithm converges to a nonoptimal solution.

There is only one example currently known for which the affine-scaling algorithm fails by converging to a nonoptimal solution. For this example, the failure occurs only for all r > 0. 995. It is not known whether there are examples of the algorithm failing for all r > 2 ∕ 3, although such a worst-case example seems likely to exist.

Convergence is only the first question. Once convergence is established, the follow-up question is: how fast? For example, given a fixed tolerance, does the affine-scaling algorithm produce a solution within this tolerance of optimality in a number of iterations that is bounded by a polynomial in n? Some variants of the path-following method have this desirable property, so one would hope that the affine-scaling method would share it. Unfortunately, while no one has written down a detailed example yet, there is strong evidence that the affine-scaling method does not have this property.

To explain the evidence, consider letting the step lengths in the affine-scaling algorithm be extremely short, even infinitesimally short. In this case, the algorithm no longer generates a sequence of points moving toward the optimal solution but rather makes a smooth curve connecting the starting point to the optimal solution. If we let the starting point vary, then we get a family of curves filling out the entire interior of the feasible region and connecting each interior point to the optimal solution. Figure 21.3 shows an example of a feasible region and some of the continuous paths. Studying the continuous paths gives information about the discrete step algorithm, since, for each point x, the step direction Δx at x is tangent to the continuous path through x. The important property that the continuous paths illustrate is that as one gets close to a face of the feasible polytope, the continuous path becomes tangent to the face (see Exercise 21.1 for an algebraic verification of this statement). This tangency holds for faces of all dimensions. In particular, it is true for edges. Hence, if one starts close to an edge, then one gets a step that looks a lot like a step of the simplex method. Therefore, it is felt that if one were to take a problem that is bad for the simplex method, such as the Klee–Minty problem, and start the affine-scaling algorithm in just the right place, then it would mimic the steps of the simplex method and therefore take 2 n iterations to get close to the optimal solution. This is the idea, but as noted above, no one has carried out the calculations.

FIGURE 21.3

A few continuous paths of the affine-scaling algorithm. At every point, the continuous path is tangent to the step direction Δx.

5 Feasibility Direction

To derive a Phase I procedure for the affine-scaling algorithm, we consider a starting point x ⁰ that has strictly positive components but does not necessarily satisfy the equality constraints Ax = b. We then let

denote the vector of infeasibilities. With these definitions under our belt, we introduce the following auxiliary problem involving one extra variable, which we shall denote by x ₀ (not to be confused with the initial solution vector x ⁰):

Clearly, the vector

is a strictly positive feasible solution to the auxiliary problem. Hence, we can apply the affine-scaling algorithm to the auxiliary problem. If the optimal solution has x ₀  ∗  > 0, then the original problem is infeasible. If, on the other hand, the optimal solution has x ₀  ∗  = 0, then the optimal solution to the auxiliary problem provides a feasible starting solution to the original problem (it may not be a strictly interior feasible solution, but we shall ignore such technicalities in the present discussion).

Let us now derive a specific formula for the step direction vector in the auxiliary problem. The vector of objective coefficients is

the constraint matrix is

and the “current” solution can be denoted as

Substituting these three objects appropriately into (21.5), we get

Exploiting heavily the fact that all the coefficients in the objective vector are zero except for the last, we can write a fairly simple expression for Δx:

The final simplification comes from applying the Sherman–Morrison–Woodbury formula (see Exercise 19.1) to the inverted expression above to discover that the vector $${(A{X}^{2}{A}^{T} + x_{0}^{2}{\rho \rho }^{T})}^{-1}\rho$$ points in the same direction as $${(A{X}^{2}{A}^{T})}^{-1}\rho$$. That is, there is a positive scalar α such that

(see Exercise 21.3). Since vector lengths are irrelevant, we can define the affine-scaling feasibility step direction as

(21.6)

6 Problems in Standard Form

We return now to problems in standard form:

Introducing slack variables w, we can write the problem equivalently as

(21.7)

Writing down the affine-scaling step direction for this problem, we get

which simplifies to

Therefore, in particular

Note that this formula for Δx matches the formula for Δx opt given in Section 19.3, except that the diagonal matrix X ² replaces XZ  − 1 and W ² replaces WY  − 1. These diagonal matrices are referred to as scaling matrices. Hence, the formula for Δx given above is often called the affine-scaling step direction with primal scaling, whereas Δx opt is referred to as the affine-scaling step direction with primal–dual scaling.

Similar connections can be established between the Phase I step direction derived in this section and Δx feas from Section 19.3. Indeed, from (21.6), we see that the feasibility step direction for (21.7) is

Again, looking just at the formula for Δx, we see that

which coincides with Δx feas except that X ² replaces XZ  − 1 and W ² replaces WY  − 1.

Exercises

21.1

Step direction becomes tangent to each facet. Let Δx denote the affine-scaling step direction given by

This step direction is clearly a function of x. Fix j. Show that the limit as x j tends to zero of Δx is a vector whose jth component vanishes. That is,

21.2

Dual Estimates. Consider the following function, defined in the interior of the polytope of feasible solutions {x : Ax = b,  x > 0} by

Consider a partition of the columns of $$A = \left [\begin{array}{cc} B &N\end{array} \right ]$$ into a basic part B and a nonbasic part N, and, as we did in our study of the simplex method, partition the n-vectors analogously. Show that

21.3

Let A be an m ×n matrix having rank m, and let ρ be an arbitrary m-vector. Use the identity proved in Exercise 19.1 to show that there exists a scalar α such that

Hint: Be mindful of which matrices are invertible.

21.4

(So-called) Dual Affine-Scaling Method. Compute the affine-scaling step-direction vector Δx for problems in the following form:

Notes

The affine-scaling algorithm was first suggested by Dikin (1967). He subsequently published a convergence analysis in Dikin (1974). Dikin’s work went largely unnoticed for many years until several researchers independently rediscovered the affine-scaling algorithm as a simple variant of Karmarkar’s algorithm (Karmarkar 1984). Of these independent rediscoveries, only two papers offered a convergence analysis: one by Barnes (1986) and the other by Vanderbei et al. (1986). It is interesting to note that Karmarkar himself was one of the independent rediscoverers, but he mistakenly believed that the algorithm enjoyed the same convergence properties as his algorithm (i.e., that it would get within any fixed tolerance of optimality within a specific number of iterations bounded by a polynomial in n).

Theorem 21.1(a) was proved by Vanderbei et al. (1986). Part (b) of the Theorem was proved by Tsuchiya and Muramatsu (1992) who also show that the result is sharp. A sharper sharpness result can be found in Hall and Vanderbei (1993). Part (c) of the Theorem was established by Mascarenhas (1997).

The first derivation of the affine-scaling feasibility step direction was given by Vanderbei (1989). The simple derivation given in Section 21.5 is due to M. Meketon.

A recent book by Saigal (1995) contains an extensive treatment of affine-scaling methods.

Bibliography

Barnes, E. (1986). A variation on Karmarkar’s algorithm for solving linear programming problems. Mathematical Programming, 36, 174–182.Crossref

Dikin, I. (1967). Iterative solution of problems of linear and quadratic programming. Soviet Mathematics Doklady, 8, 674–675.

Dikin, I. (1974). On the speed of an iterative process. Upravlyaemye Sistemi, 12, 54–60.

Hall, L., and Vanderbei, R. (1993). Two-thirds is sharp for affine scaling. OR Letters, 13, 197–201.

Karmarkar, N. (1984). A new polynomial time algorithm for linear programming. Combinatorica, 4, 373–395.Crossref

Mascarenhas, W. (1997). The affine scaling algorithm fails for λ = 0. 999. SIAM Journal on Optimization, 7, 34–46.

Saigal, R. (1995). Linear programming. Boston: Kluwer.Crossref

Tsuchiya, T., and Muramatsu, M. (1992). Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems. SIAM Journal on Optimization, 5(3), 525–551.Crossref

Vanderbei, R. (1989). Affine scaling for linear programs with free variables. Mathematical Programming, 43, 31–44.Crossref

Vanderbei, R., Meketon, M., and Freedman, B. (1986). A modification of Karmarkar’s linear programming algorithm. Algorithmica, 1, 395–407.Crossref

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_22

22. The Homogeneous Self-Dual Method

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In Chapter 18, we described and analyzed an interior-point method called the path-following algorithm. This algorithm is essentially what one implements in practice but as we saw in the section on convergence analysis, it is not easy (and perhaps not possible) to give a complete proof that the method converges to an optimal solution. If convergence were completely established, the question would still remain as to how fast is the convergence. In this chapter, we shall present a similar algorithm for which a complete convergence analysis can be given.

1 From Standard Form to Self-Dual Form

As always, we are interested in a linear programming problem given in standard form

(22.1)

and its dual

(22.2)

As we shall show, these two problems can be solved by solving the following problem, which essentially combines the primal and dual problems into one problem:

(22.3)

Note that, beyond combining the primal and dual into one big problem, one new variable (ϕ) and one new constraint have been added. Hence, the total number of variables in (22.3) is $$n + m + 1$$ and the total number of constraints is $$n + m + 1$$. Furthermore, the objective function and the right-hand sides all vanish. Problems with such right-hand sides are called homogeneous. Also, the constraint matrix for problem (22.3) is skew symmetric. That is, it is equal to the negative of its transpose. Homogeneous linear programming problems having a skew symmetric constraint matrix are called self-dual.

In the next section, we shall give an algorithm for the solution of homogeneous self-dual linear programming problems. But first, let’s note that a solution to (22.3) in which ϕ > 0 can be converted into solutions for (22.1) and (22.2). Indeed, let $$(\bar{x},\bar{y},\bar{\phi })$$ be an optimal solution to problem (22.3). Suppose that $$\bar{\phi }> 0$$. (The algorithm given in the next section will guarantee that this property is satisfied whenever (22.1) and (22.2) have optimal solutions.¹) Put

Then the constraints in (22.3) say that

Also, x  ∗  and y  ∗  are both nonnegative. Therefore, x  ∗  is feasible for (22.1) and y  ∗  is feasible for (22.2). From the weak duality theorem together with the third inequality above, we get

Therefore, x  ∗  is optimal for the primal problem (22.1) and y  ∗  is optimal for the dual problem (22.2). As we will see later, the case where $$\bar{\phi }= 0$$ corresponds to infeasibility of either the primal or the dual problem (or both).

2 Homogeneous Self-Dual Problems

Consider a linear programming problem in standard form

and its dual

Such a linear programming problem is called self-dual if m = n, $$A = -{A}^{T}$$, and $$b = -c$$. The reason for the name is that the dual of such a problem is the same as the primal. To see this, rewrite the constraints as less-thans and then use the defining properties for self-duality to get

Similarly, writing the objective function as a maximization, we get

Hence, ignoring the (irrelevant) fact that the dual records the negative of the objective function, the primal and the dual are seen to be the same. A linear programming problem in which the right-hand side vanishes is called a homogeneous problem. It follows that if a problem is homogeneous and self-dual, then its objective function must vanish too.

For the remainder of this section, we assume that the problem under consideration is homogeneous and self-dual. Since the case $$m = n = 1$$ is trivial (A = 0 in this case), we assume throughout this section that n ≥ 2. Also, since the dual is the same problem as the primal, we prefer to use the letter z for the primal slacks (instead of the usual w). Hence, the primal can be written as

(22.4)

The following theorem establishes some of the important properties of homogeneous self-dual problems.

Theorem 22.1.

For homogeneous self-dual problem  (22.4) , the following statements hold:

1.

It has feasible solutions and every feasible solution is optimal.

2.

The set of feasible solutions has empty interior. In fact, if (x,z) is feasible, then z T x = 0.

Proof.

(1) The trivial solution, (x, z) = (0, 0), is feasible. Since the objective function is zero, every feasible solution is optimal.

(2) Suppose that (x, z) is feasible for (22.4). The fact that A is skew symmetric implies that ξ T Aξ = 0 for every vector ξ (see Exercise 16.​1). In particular, x T Ax = 0. Therefore, multiplying $$Ax + z = 0$$ on the left by x T , we get $$0 = {x}^{T}Ax + {x}^{T}z = {x}^{T}z$$. This completes the proof. □ 

Part (2) of the previous Theorem tells us that homogeneous self-dual problems do not have central paths.

2.1 Step Directions

As usual, the interior-point method we shall derive will have the property that the intermediate solutions it produces will be infeasible. Hence, let

denote the infeasibility of a solution (x, z). Also, let

The number μ(x, z) measures the degree of noncomplementarity between x and z. When x and z are clear from context, we shall simply write ρ for ρ(x, z) and μ for μ(x, z).

Step directions (Δx, Δz) are chosen to reduce the infeasibility and noncomplementarity of the current solution by a given factor δ, 0 ≤ δ ≤ 1. Hence, we consider the nonlinear system that would make the infeasibility and noncomplementarity of $$(x + \Delta x,z + \Delta z)$$ be δ times that of (x, z):

As usual, this system is nonlinear in the “delta” variables. Dropping the nonlinear term (appearing only in the second equation), we get the following linear system of equations for the step directions:

(22.5)

(22.6)

With these step directions, we pick a step length θ and step to a new point:

We denote the new ρ-vector by $$\bar{\rho }$$ and the new μ-value by $$\bar{\mu }$$:

The following theorem establishes some of the properties of these step directions.

Theorem 22.2.

The following relations hold:

1.

Δz T Δx = 0.

2.

$$\bar{\rho }= (1 -\theta +\theta \delta )\rho$$.

3.

$$\bar{\mu }= (1 -\theta +\theta \delta )\mu$$.

4.

$$\bar{X}\bar{Z}e -\bar{\mu } e = (1-\theta )(XZe -\mu e) {+\theta }^{2}\Delta X\Delta Ze$$.

Proof.

(1) We start by multiplying both sides of (22.5) on the left by Δx T :

(22.7)

The skew symmetry of A (i.e., $$A = -{A}^{T}$$) implies that Δx T AΔx = 0 (see Exercise 16.​1). Hence, the left-hand side of (22.7) simplifies nicely:

Substituting the definition of ρ into the right-hand side of (22.7), we get

Next, we use the skew symmetry of A to rewrite Δx T Ax as follows:

Assembling what we have so far, we see that

(22.8)

To proceed, we use (22.5) to replace AΔx with $$-(1-\delta )\rho - \Delta z$$. Therefore,

(22.9)

Again using the definition of ρ and the skew symmetry of A, we see that

The last two terms in (22.9) can be simplified by multiplying both sides of (22.6) on the left by e T and then using the definition of μ to see that

Making these substitutions in (22.9), we get

Hence, from (22.8), we see that Δx T Δz vanishes as claimed.

(2) From the definitions of $$\bar{x}$$ and $$\bar{z}$$, we see that

(3) From the definitions of $$\bar{x}$$ and $$\bar{z}$$, we see that

From part (1) and (22.6), we then get

Therefore,

(4) From the definitions of $$\bar{x}$$ and $$\bar{z}$$ together with part (3), we see that

Substituting (22.6) into the second term on the right and recollecting terms, we get the desired expression. □ 

2.2 Predictor-Corrector Algorithm

With the preliminaries behind us, we are now ready to describe an algorithm. We shall be more conservative than we were in Chapter 18 and define the algorithm in such a way that it keeps the components of XZe close to each other. Indeed, for each 0 ≤ β ≤ 1, let

Shortly, we will only deal with $$\mathcal{N}(1/4)$$ and $$\mathcal{N}(1/2)$$ but first let us note generally that β < β′ implies that $$\mathcal{N}(\beta ) \subset \mathcal{N}(\beta ^{\prime})$$. Hence, as a function of β, the $$\mathcal{N}(\beta )$$’s form an increasing family of sets. Also, $$\mathcal{N}(0)$$ is precisely the set of points (x, z) for which XZe has all equal components.

The algorithm alternates between two types of steps. On the first iteration and subsequently on every other iteration, the algorithm performs a predictor step. Before a predictor step, one assumes that

Then step directions are computed using δ = 0 (i.e., with no centering) and the step length is calculated so as not to go outside of $$\mathcal{N}(1/2)$$:

(22.10)

On the even iterations, the algorithm performs a corrector step. Before a corrector step, one assumes that

(as is guaranteed by the predictor step’s step length). Then step directions are computed using δ = 1 (i.e., pure centering) and the step length parameter θ is set to 1.

The following theorem shows that the result of each step satisfies the precondition for the next step of the algorithm and that μ decreases on predictor steps while it stays the same on corrector steps.

Theorem 22.3.

The following statements are true:

1.

After a predictor step, $$(\bar{x},\bar{z}) \in \mathcal{N}(1/2)$$ and $$\bar{\mu }= (1-\theta )\mu$$.

2.

After a corrector step, $$(\bar{x},\bar{z}) \in \mathcal{N}(1/4)$$ and $$\bar{\mu }=\mu$$.

Proof of Part (1). 

The formula for $$\bar{\mu }$$ follows from part (3) of Theorem 22.2 by putting δ = 0. The fact that $$(\bar{x},\bar{z}) \in \mathcal{N}(1/2)$$ is an immediate consequence of the choice of θ. □ 

Before proving Part (2) of the Theorem, we need to introduce some notation and prove a few technical results. Let

(22.11)

The technical results are summarized in the following lemma.

Lemma 22.4.

The following statements are true:

1.

$$\|PQe\| \leq \frac{1} {2}\|{r\|}^{2}$$.

2.

If δ = 0, then $$\|{r\|}^{2} = n\mu$$.

3.

If δ = 1 and $$(x,z) \in \mathcal{N}(\beta )$$ , then $$\|{r\|}^{2} {\leq \beta }^{2}\mu /(1-\beta )$$.

Proof.

(1) First note that $${p}^{T}q = \Delta {x}^{T}\Delta z = 0$$ by Theorem 22.2(1). Hence,

Therefore,

Taking square roots yields the desired inequality.

(2) Putting δ = 0 in (22.11), we see that $$r = -{X}^{1/2}{Z}^{1/2}e$$. Therefore, $$\|{r\|}^{2} = {z}^{T}x = n\mu$$.

(3) Suppose that $$(x,z) \in \mathcal{N}(\beta )$$. Whenever the norm of a vector is smaller than some number, the magnitude of each component of the vector must also be smaller than this number. Hence, | x j z j  − μ | ≤ βμ. It is easy to see that this inequality is equivalent to

(22.12)

Now putting δ = 1 in (22.11), we get

Therefore, using the lower bound given in (22.12), we get the following upper bound:

Finally, since $$(x,z) \in \mathcal{N}(\beta )$$, we see that the above sum is bounded by β ² μ ². This gives the claimed inequality. □ 

Proof of Theorem 22.3(2). 

Since θ = 1 in a corrector step, it follows from Theorem 22.2(4) that $$\bar{X}\bar{Z}e -\bar{\mu } e = \Delta X\Delta Ze = PQe$$. Therefore, parts (1) and (3) of Lemma 22.4 imply that

(22.13)

We also need to show that $$(\bar{x},\bar{z}) > 0$$. For 0 ≤ t ≤ 1, let

Then from part (4) of Theorem 22.2, we have

The right-hand side is the sum of two vectors. Since the length of the sum of two vectors is less than the sum of the lengths (i.e., by the triangle inequality), it follows that

(22.14)

(note that we’ve pulled the scalars out of the norms). Now, since $$(x,z) \in \mathcal{N}(1/2)$$, we have $$\|XZe -\mu e\| \leq \mu /2$$. Furthermore, from (22.13) we have that $$\|\Delta X\Delta Ze\| =\| PQe\| \leq \mu /4$$. Replacing the norms in (22.14) with these upper bounds, we get the following bound:

(22.15)

(the second inequality follows from the obvious facts that t ² ≤ t and $$\mu /4 \leq \mu /2$$).

Now, consider a specific component j. It follows from (22.15) that

Since δ = 1, part (3) of Theorem 22.2 tells us that μ(t) = μ for all t. Therefore the previous inequality can be written as

(22.16)

This inequality then implies that x j (t) > 0 and z j (t) > 0 for all 0 ≤ t ≤ 1 (since they could only become negative by passing through 0, which is ruled out by (22.16)). Putting t = 1, we get that $$\bar{x}_{j} > 0$$ and $$\bar{z}_{j} > 0$$. Since the component j was arbitrary, it follows that $$(\bar{x},\bar{z}) > 0$$. Therefore $$(\bar{x},\bar{z}) \in \mathcal{N}(1/4)$$. □ 

2.3 Convergence Analysis

The previous theorem showed that the predictor-corrector algorithm is well defined. The next theorem gives us a lower bound on the progress made by each predictor step.

Theorem 22.5.

In each predictor step, $$\theta \geq \frac{1} {2\sqrt{n}}$$.

Proof.

Using the same notation as in the proof of Theorem 22.3, we have the inequality:

(22.17)

This time, however, $$(x,z) \in \mathcal{N}(1/4)$$ and δ = 0. Hence,

and, from parts (1) and (2) of Lemma 22.4,

Using these two bounds in (22.17), we get the following bound:

Now, fix a $$t \leq {(2\sqrt{n})}^{-1}$$. For such a t, we have $${t}^{2}n/2 \leq 1/8$$. Therefore, using the fact that t ≤ 1 ∕ 2 for n ≥ 2, we get

Hence, as in the previous theorem, $$(x(t),z(t)) \in \mathcal{N}(1/2)$$. Since t was an arbitrary number less than $${(2\sqrt{n})}^{-1}$$, it follows that $$\theta \geq {(2\sqrt{n})}^{-1}$$. □ 

Let (x (k), z (k)) denote the solution after the kth iteration and let

The algorithm starts with $${x}^{(0)} = {z}^{(0)} = e$$. Therefore, μ ⁽⁰⁾ = 1. Our aim is to show that μ (k) and ρ (k) tend to zero as k tends to infinity. The previous theorem together with Theorem 22.3 implies that, after an even number of iterations, say 2k, the following inequality holds:

Also, since the corrector steps don’t change the value of μ, it follows that

From these two statements, we see that

Now, consider ρ (k). It follows from parts (2) and (3) of Theorem 22.2 that the reduction in infeasibility tracks the reduction in noncomplementarity. Hence,

Therefore, the fact that μ (k) tends to zero implies the same for ρ (k).

In fact, more can be said:

Theorem 22.6.

The limits $${x}^{{\ast}} =\lim _{k\rightarrow \infty }{x}^{(k)}$$ and $${z}^{{\ast}} =\lim _{k\rightarrow \infty }{z}^{(k)}$$ exist and (x ∗ ,z ∗ ) is optimal. Furthermore, the vectors x ∗ and z ∗ are strictly complementary to each other. That is, for each j, $$x_{j}^{{\ast}}z_{j}^{{\ast}} = 0$$ but either x j ∗ > 0 or z j ∗ > 0.

The proof is fairly technical, and so instead of proving it, we prove the following theorem, which captures the main idea.

Theorem 22.7.

There exist positive constants $$c_{1},c_{2},\ldots,c_{n}$$ such that $$(x,z) \in \mathcal{N}(\beta )$$ implies that $$x_{j} + z_{j} \geq c_{j} > 0$$ for each j = 1,2,…,n.

Proof.

Put μ = μ(x, z) and $$\rho =\rho (x,z) {=\mu \rho }^{(0)}$$. Let (x  ∗ , z  ∗ ) be a strictly complementary feasible solution (the existence of which is guaranteed by Theorem 10.5). We begin by studying the expression $${z}^{T}{x}^{{\ast}} + {x}^{T}{z}^{{\ast}}$$. Since $$A{x}^{{\ast}} + {z}^{{\ast}} = 0$$, we have that

By the skew-symmetry of A, we see that $$-{A}^{T}x + z = Ax + z =\rho$$. And, since ρ = μρ ⁽⁰⁾, we get

(22.18)

The factor $${{\rho }^{(0)}}^{T}{x}^{{\ast}}$$ is a constant (i.e., it does not depend on x or z). Let us denote it by M. Since all the terms in the two products on the left in (22.18) are nonnegative, it follows that each one is bounded by the right-hand side. So if we focus on a particular index j, we get the following bounds:

(22.19)

Now, we use the assumption that $$(x,z) \in \mathcal{N}(\beta )$$ to see that

In other words, $$\mu \leq x_{j}z_{j}/(1-\beta )$$, and so the inequalities in (22.19) become

Since x j and z j are strictly positive, we can divide by them (and the constants) to get

Putting

we get the desired lower bound on x j  + z j . □ 

2.4 Complexity of the Predictor-Corrector Algorithm

Of course, in practice we don’t run an infinite number of iterations. Instead, we set a priori a threshold and stop when μ (k) falls below it. The threshold is usually denoted by 2 − L where L is some number. Typically, we want the threshold to be about 10 − 8, which corresponds to L ≈ 26.

As we saw before, after an even number of iterations, say 2k, the μ-value is bounded by the following inequality:

Hence, it suffices to pick a k big enough to have

Taking logarithms of both sides and solving for k, we see that any

will do. Since $$-\log (1 - x) \geq x$$, we get

Therefore, any $$k \geq 2L\sqrt{n}$$ will do. In particular, $$k = 2L\sqrt{n}$$ rounded up to the nearest integer will suffice. Since k represents half the number of iterations, it follows that it will take at most $$4L\sqrt{n}$$ iterations for the μ-value to fall below the threshold of 2 − L. This bound implies that the method is a polynomial algorithm, since it says that any desired precision can be obtained in a number of iterations that is bounded above by a polynomial in n (here, $$4L\sqrt{n}$$ is not itself a polynomial but is bounded above by say a linear function in n for n ≥ 2).

2.5 The KKT System

We end this section on homogeneous self-dual problems by briefly discussing the KKT system (22.5)–(22.6). Solving this system of equations is the most time consuming step within each iteration of the predictor-corrector algorithm. There are several ways in which one can organize the computation. The approach that most parallels what we have done before is first to solve (22.6) for Δz,

(22.20)

and then to eliminate it from (22.5) to get the following reduced KKT system:

In the next section, we apply the algorithm developed in this section to the homogeneous self-dual problem given by (22.3).

3 Back to Standard Form

We return now to the setup in Section 1. Let z, w, and ψ denote the slack variables for the constraints in problem (22.3):

(22.21)

We say that a feasible solution $$(\bar{x},\bar{y},\bar{\phi },\bar{z},\bar{w},\bar{\psi })$$ is strictly complementary if $$\bar{x}_{j} +\bar{ z}_{j} > 0$$ for all j, $$\bar{y}_{i} +\bar{ w}_{i} > 0$$ for all i, and $$\bar{\phi }+\bar{\psi } > 0$$. Theorem 10.5 ensures the existence of such a solution (why?).

The following theorem summarizes and extends the motivating discussion given in Section 1.

Theorem 22.8.

Suppose that $$(\bar{x},\bar{y},\bar{\phi },\bar{z},\bar{w},\bar{\psi })$$ is a strictly complementary feasible (hence, optimal) solution to (22.21).

1.

If $$\bar{\phi }> 0$$ , then $${x}^{{\ast}} =\bar{ x}/\bar{\phi }$$ is optimal for the primal problem (22.1) and $${y}^{{\ast}} =\bar{ y}/\bar{\phi }$$ is optimal for its dual (22.2).

2.

If $$\bar{\phi }= 0$$ , then either $${c}^{T}\bar{x} > 0$$ or $${b}^{T}\bar{y} < 0$$.

a.

If $${c}^{T}\bar{x} > 0$$ , then the dual problem is infeasible.

b.

If $${b}^{T}\bar{y} < 0$$ , then the primal problem is infeasible.

Proof.

Part (1) was proved in Section 1. For part (2), suppose that $$\bar{\phi }= 0$$. By strict complementarity, $$\bar{\psi }> 0$$. Hence, $$\bar{x}$$ and $$\bar{y}$$ satisfy

(22.22)

From the last inequality, we see that it is impossible to have $${b}^{T}\bar{y} \geq 0$$ and $${c}^{T}\bar{x} \leq 0$$. That is, either $${c}^{T}\bar{x} > 0$$ or $${b}^{T}\bar{y} < 0$$ (or both). Suppose, without loss of generality, that $${c}^{T}\bar{x} > 0$$. We will prove by contradiction that the dual problem is infeasible. To this end, suppose that there exists a vector y ⁰ ≥ 0 such that

(22.23)

Since $$\bar{x} \geq 0$$, we can multiply by it without changing the direction of an inequality. So multiplying (22.23) on the left by $$\bar{{x}}^{T}$$, we get

Now, the right-hand side is strictly positive. But inequality (22.22) together with the nonnegativity of y ⁰ implies that the left-hand side is nonpositive:

This is a contradiction and therefore the dual must be infeasible. □ 

3.1 The Reduced KKT System

The right-hand side in the reduced KKT system involves the vector of infeasibilities. We partition this vector into three parts as follows:

The reduced KKT system for (22.3) is given by

(22.24)

where

This system is not symmetric. One could use a general purpose equation solver to solve it, but its special structure would be mostly ignored by such a solver. To exploit the structure, we solve this system in two stages. We start by using the first two equations to solve simultaneously for Δx and Δy in terms of Δϕ:

Introducing abbreviating notations, we can write

(22.25)

where the vectors

are found by solving the following two systems of equations:

and

Then we use (22.25) to eliminate Δx and Δy from the last equation in (22.24):

We then solve for Δϕ:

Given Δϕ, (22.25) determines Δx and Δy. Once these vectors are known, (22.20) is used to compute the step directions for the slack variables:

We now see that the reduced KKT system can be solved by solving two systems of equations for f and g. These two systems both involve the same matrix. Furthermore, these systems can be formulated as quasidefinite systems by negating the first equation and then reordering the equations appropriately. For example, the quasidefinite system for g is

Therefore, the techniques developed in Chapter 20 can be used to solve these systems of equations. In particular, to solve the two systems, the quasidefinite matrix only needs to be factored once. Then the two systems can be solved by doing two forward and two backward substitutions. Since factorization requires more computation than forward and backward substitutions, one would expect to be able to solve these two systems in much less time than if they were each being solved from scratch. In fact, it is often the case that one can solve two systems involving the same quasidefinite matrix in little more time than is required to solve just one such system.

The full homogeneous self-dual method is summarized in Figure 22.1.

FIGURE 22.1

The homogeneous self-dual method.

4 Simplex Method vs. Interior-Point Methods

Finally, we compare the performance of interior-point methods with the simplex method. For this comparison, we have chosen the homogeneous self-dual method described in this chapter and the self-dual simplex method (see Figure 7.​1). In the interest of efficiency certain liberties have been taken with the implementations. For example, in the homogeneous self-dual method, (18.​6) is used to compute “long” step lengths instead of the more conservative “short” step lengths in (22.10). The code fragments implementing each of these two algorithms are shown in Appendix A.

A standard collection of test problems, the so-called NETLIB suite, were used in the comparison. Problems in this collection are formulated with bounds and ranges:

However, to keep the algorithms as simple as possible, they were implemented only for problems in our standard inequality form. Therefore, the problems from the NETLIB suite were converted to standard form as follows:

Of course, this transformation is invalid when any of the lower bounds are infinite. Therefore, such problems have been dropped in our experiment. Also, this transformation introduces significant computational inefficiency but, since it was applied equally to the problems presented to both methods, the comparison remains valid.

The results of our experiment are shown in Table 22.1. The most obvious observation is that the simplex method is generally faster and that, for many problems, the slower method is not more than 3 or 4 times slower. For problems in this suite, these results are consistent with results reported in the literature. However, it must be noted that the problems in this suite range only from small to medium in size. The largest problem, fit2p, has about 3,000 constraints and about 14,000 variables. By today’s standards, this problem is considered of medium size. For larger problems, reports in the literature indicate that interior point methods tend to be superior although the results are very much dependent on the specific class of problems. In the remaining chapters of this book we shall consider various extensions of the linear programming model. We shall see that the simplex method is particularly well suited for solving integer programming problems studied in Chapter 23 whereas interior point methods are more appropriate for extensions into the quadratic and convex programming problems studied in Chapters 24 and 25. These considerations are often more important than speed. There are, of course, exceptions. For example, the interior-point method is about 900 times faster than the simplex method on problem fit2p. Such a difference cannot be ignored.

Table 22.1

Comparison between the self-dual simplex method and the homogeneous self-dual interior-point method.

Name

Time

Name

Time

Simplex

Interior

Simplex

Interior

method

point

method

point

25fv47

2 min 55.70 s

3 min 14.82 s

maros

1 min 0.87 s

3 min 19.43 s

80bau3b

7 min 59.57 s

2 min 34.84 s

nesm

1 min 40.78 s

6 min 21.28 s

adlittle

0 min 0.26 s

0 min 0.47 s

pilot87

*

*

afiro

0 min 0.03 s

0 min 0.11 s

pilotnov

*

4 min 15.31 s

agg

0 min 1.09 s

0 min 4.59 s

pilots

*

32 min 48.15 s

agg2

0 min 1.64 s

0 min 21.42 s

recipe

0 min 0.21 s

0 min 1.04 s

agg3

0 min 1.72 s

0 min 26.52 s

sc105

0 min 0.28 s

0 min 0.37 s

bandm

0 min 15.87 s

0 min 9.01 s

sc205

0 min 1.30 s

0 min 0.84 s

beaconfd

0 min 0.67 s

0 min 6.42 s

sc50a

0 min 0.09 s

0 min 0.17 s

blend

0 min 0.40 s

0 min 0.56 s

sc50b

0 min 0.12 s

0 min 0.15 s

bnl1

0 min 38.38 s

0 min 46.09 s

scagr25

0 min 12.93 s

0 min 4.44 s

bnl2

3 min 54.52 s

10 min 19.04 s

scagr7

0 min 1.16 s

0 min 1.05 s

boeing1

0 min 5.56 s

0 min 9.14 s

scfxm1

0 min 4.44 s

0 min 7.80 s

boeing2

0 min 0.80 s

0 min 1.72 s

scfxm2

0 min 14.33 s

0 min 18.84 s

bore3d

0 min 1.17 s

0 min 3.97 s

scfxm3

0 min 28.92 s

0 min 28.92 s

brandy

0 min 5.33 s

0 min 8.44 s

scorpion

0 min 3.38 s

0 min 2.64 s

czprob

0 min 50.14 s

0 min 41.77 s

scrs8

0 min 7.15 s

0 min 9.53 s

d2q06c

*

1 h 11 min 1.93 s

scsd1

0 min 0.86 s

0 min 3.88 s

d6cube

2 min 46.71 s

13 min 44.52 s

scsd6

0 min 2.89 s

0 min 9.31 s

degen2

0 min 17.28 s

0 min 17.02 s

scsd8

0 min 28.87 s

0 min 16.82 s

degen3

5 min 55.52 s

3 min 36.73 s

sctap1

0 min 2.98 s

0 min 3.08 s

dfl001

8 h 55 min 33.05 s

**

sctap2

0 min 7.41 s

0 min 12.03 s

e226

0 min 4.76 s

0 min 6.65 s

sctap3

0 min 11.70 s

0 min 17.18 s

etamacro

0 min 17.94 s

0 min 43.40 s

seba

0 min 27.25 s

0 min 11.90 s

fffff800

0 min 10.07 s

1 min 9.15 s

share1b

0 min 2.07 s

0 min 10.90 s

finnis

0 min 4.76 s

0 min 6.17 s

share2b

0 min 0.47 s

0 min 0.71 s

fit1d

0 min 18.15 s

0 min 11.63 s

shell

0 min 16.12 s

0 min 29.45 s

fit1p

7 min 10.86 s

0 min 16.47 s

ship04l

0 min 3.82 s

0 min 13.60 s

fit2d

1 h 3 min 14.37 s

4 min 27.66 s

ship04s

0 min 3.48 s

0 min 10.81 s

fit2p

36 h 31 min 31.80 s

2 min 35.67 s

ship08l

0 min 17.83 s

0 min 39.06 s

forplan

0 min 3.99 s

*

ship08s

0 min 8.85 s

0 min 19.64 s

ganges

0 min 44.27 s

0 min 34.89 s

ship12l

0 min 26.55 s

1 min 8.62 s

gfrdpnc

0 min 11.51 s

0 min 8.46 s

ship12s

0 min 16.75 s

0 min 30.33 s

greenbea

22 min 45.49 s

43 min 4.32 s

sierra

0 min 10.88 s

0 min 42.89 s

grow15

0 min 8.55 s

0 min 58.26 s

standata

0 min 0.57 s

0 min 6.60 s

grow22

0 min 11.79 s

2 min 0.53 s

standmps

0 min 2.41 s

0 min 13.44 s

grow7

0 min 3.61 s

0 min 13.57 s

stocfor1

0 min 0.22 s

0 min 0.92 s

israel

0 min 1.83 s

0 min 2.66 s

stocfor2

0 min 45.15 s

0 min 40.43 s

kb2

0 min 0.15 s

0 min 0.34 s

wood1p

0 min 14.15 s

7 min 18.47 s

lotfi

0 min 0.81 s

0 min 3.36 s

woodw

1 min 48.14 s

8 min 53.92 s

maros-r7

*

1 h 31m12.06 s

(*) Denotes numerical difficulties (**) Denotes insufficient memory

When comparing algorithms it is always tempting to look for ways to improve the slower method. There are obvious enhancements to the interior-point method used in this implementation. For example, one could use the same LDL T factorization to compute both the predictor and the corrector directions. When implemented properly, this enhancement alone can almost halve the times for this method.

Of course, the winning algorithm can also be improved (but, significant overall improvements such as the one just mentioned for the interior-point method are not at all obvious). Looking at the table, we note that the interior-point method solved both fit2p and fit2d in roughly the same amount of time. These two problems are duals of each other and hence any algorithm that treats the primal and the dual symmetrically should take about the same time to solve them. Now, look at the simplex method’s performance on these two problems. There is a factor of 36 difference between them. The reason is that, even though we have religiously adhered to primal-dual symmetry in our development of the simplex method, an asymmetry did creep in. To see it, note that the basic matrix is always a square submatrix of $$\left [\begin{array}{cc} A&I \end{array} \right ]$$. That is, it is an m ×m matrix. If we apply the algorithm to the dual problem, then the basis matrix is n ×n. Hence, even though the sequence of iterates generated should be identical with the two problems, the computations involved in each iteration can be very different if m and n are not about the same. This is the case for the fit2p/fit2d pair. Of course, one can easily think up schemes to overcome this difficulty. But even if the performance of the simplex method on fit2p can be brought in line with its performance on fit2d, it will still be about 25 times slower than the interior-point on this problem—a difference that remains significant.

Exercises

22.1

When n = 1, the set $$\mathcal{N}(\beta )$$ is a subset of ℝ ². Graph it.

22.2

Suppose there is an algorithm for which one can prove that

for every k ≥ 1, where f(n) denotes a specific function of n, such as f(n) = n ², and a is a constant. In terms of a and f and the “precision” L, give a (tight) upper bound on the number of iterations that would be sufficient to guarantee that

22.3

In Section 3 of Chapter 20, we extended the primal-dual path-following method to treat problems in general form. Extend the homogeneous self-dual method in the same way.

22.4

Long-step variant. Let

(The notation minXZe denotes the scalar that is the minimum of all the components of the vector XZe. Throughout this problem, given any vector v, the notation minv (maxv) will denote the minimum (maximum) of the components of v.) Fix $$\frac{1} {2} <\beta < 1$$ (say β = 0. 95). A long-step variant of the homogeneous self-dual method starts with an initial $$(x,z) \in \mathcal{M}(\beta )$$ and in every iteration uses

and

The goal of this exercise is to analyze the complexity of this algorithm by completing the following steps.

(a)

Show that $$\mathcal{N}(\beta ) \subset \mathcal{M}(\beta ) \subset \mathcal{M}(1) =\{ (x,z) > 0\}$$.

(b)

Show that $$\max (-PQe) \leq \| {r\|}^{2}/4$$. Hint: Start by writing

and then use the facts that p T q = 0, $$p_{i} + q_{i} = r_{i}$$ , and that for any two real numbers a and b, (a + b) ² ≥ 4ab (prove this).

(c)

Show that if $$(x,z) \in \mathcal{M}(\beta )$$, then $$\|{r\|}^{2} \leq n\mu$$. Hint: Use (22.11) to write $$\|{r\|}^{2} =\sum _{j}{(x_{j}z_{j}-\delta \mu )}^{2}/x_{j}z_{j}$$ . Expand the numerator, use the definitions of μ and δ to simplify, and then use the assumption that $$(x,z) \in \mathcal{M}(\beta )$$ to take care of the remaining denominator.

(d)

Show that if $$(x,z) \in \mathcal{M}(\beta )$$, then

Hint: Using the same notation as in the proof of Theorem  22.3 , fix $$t \leq \min \{ 1,-\beta \delta \mu /\min PQe\}$$ , write

and then replace the right-hand side by a lower bound that is independent of j. From there, follow your nose until you get the first inequality. The second inequality follows from parts (b) and (c) .

(e)

As usual letting μ (k) denote the μ value associated with the solution on the kth iteration of the algorithm, show that

(f)

Give an upper bound on the number of iterations required to get μ (k) ≤ 2 − L.

(g)

Show that θ can be computed by solving n (univariate) quadratic equations.

(h)

A robust implementation of a quadratic equation solver uses the formula

for one of the two roots to $$a{x}^{2} + bx + c = 0$$ (a similar formula is used for the other one). Show that the two expressions on the right are mathematically equal and suggest a reason to prefer one over the other in the particular cases indicated.

Notes

The first study of homogeneous self-dual problems appeared in Tucker (1956). This chapter is based on the papers Mizuno et al. (1993), Ye et al. (1994), and Xu et al. (1993). The step length formula (22.10) forces the algorithm studied in this chapter to take much shorter steps than those in Chapter 18. In general, algorithms that are based on steps that confine the iterates to $$\mathcal{N}(\beta )$$ are called short-step methods. A long-step variant of the algorithm can be obtained by enlarging the set $$\mathcal{N}(\beta )$$. Such a variant is the subject of Exercise 22.4. For this method, a worst case analysis shows that it takes on the order of n steps to achieve a given level of precision. Xu et al. (1993) describes an efficient implementation of the long-step variant.

The predictor–corrector method is a standard technique used in the numerical solution of ordinary differential equations. Mehrotra (1992) (see also Mehrotra 1989) was the first to apply this technique in the context of interior-point methods, although the related notion of forming power series approximations was suggested earlier by N.K. Karmarkar and is described in Adler et al. (1989).

Bibliography

Adler, I., Karmarkar, N., Resende, M., and Veiga, G. (1989). An implementation of Karmarkar’s algorithm for linear programming. Mathematical Programming, 44, 297–335.Crossref

Mehrotra, S. (1989). Higher order methods and their performance (Techincal Report TR 90-16R1) Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston. Revised July 1991.

Mehrotra, S. (1992). On the implementation of a (primal-dual) interior point method. SIAM Journal on Optimization, 2, 575–601.Crossref

Mizuno, S., Todd, M., and Ye, Y. (1993). On adaptive-step primal-dual interior-point algorithms for linear programming. Mathematics of Operations Research, 18, 964–981.Crossref

Tucker, A. (1956). Dual systems of homogeneous linear equations. Annals of Mathematics Studies, 38, 3–18.

Xu, X., Hung, P., and Ye, Y. (1993). A simplified homogeneous and self-dual linear programming algorithm and its implementation (Techincal Report). College of Business Administration, University of Iowa. To appear in Annals of Operations Research.

Ye, Y., Todd, M., and Mizuno, S. (1994). An $$o(\sqrt{n}l)$$-iteration homogeneous and self-dual linear programming algorithm. Mathematics of Operations Research, 19, 53–67.Crossref

Footnotes

1

The astute reader might notice that setting all variables to 0 produces an optimal solution.

Part 4
Extensions

It’s hard. But it’s harder to ignore it. —C. Stevens

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_23

23. Integer Programming

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

Many real-world problems could be modeled as linear programs except that some or all of the variables are constrained to be integers. Such problems are called integer programming problems. One might think that these problems wouldn’t be much harder than linear programming problems. For example, we saw in Chapter 14 that for network flow problems with integer data, the simplex method automatically produces integer solutions. But that was just luck. In general, one can’t expect to get integer solutions; in fact, as we shall see in this chapter, integer programming problems turn out to be generally much harder to crack than linear ones.

There are many important real-world problems that can be formulated as integer programming problems. The subject is so important that several monographs are devoted entirely to it. In this chapter, we shall just present a few favorite applications that can be modeled as integer programming problems and then we will discuss one technique for solving problems in this class, called branch-and-bound.

1 Scheduling Problems

There are many problems that can be classified as scheduling problems. We shall consider just two related problems of this type: the equipment scheduling and crew scheduling problems faced by large airlines. Airlines determine how to route their planes as follows. First, a number of specific flight legs are defined based on market demand. A leg is by definition one flight taking off from somewhere at some time and landing somewhere else (hopefully). For example, a leg could be a flight from New York directly to Chicago departing at 7:30 a.m. Another might be a flight from Chicago to San Francisco departing at 1:00 p.m. The important point is that these legs are defined by market demand, and it is therefore not clear a priori how to put these legs together in such a way that the available aircraft can cover all of them. That is, for each airplane, the airline must put together a route that it will fly. A route, by definition, consists of a sequence of flight legs for which the destination of one leg is the origin of the next (and, of course, the final destination must be the origin of the first leg, forming a closed loop).

The airline scheduling problems are generally tackled in two stages. First, reasonable routes are identified that meet various regulatory and temporal constraints (you can’t leave somewhere before you’ve arrived there—time also must be reserved for dropping off and taking on passengers). This route-identification problem is by no means trivial, but it isn’t our main interest here, so we shall simply assume that a collection of reasonable routes has already been identified. Given the potential routes, the second stage is to select a subset of them with the property that each leg is covered by exactly one route. If the collection of potential routes is sufficiently rich, we would expect there to be several feasible solutions. Therefore, as always, our goal is to pick an optimal one, which in this case we define as one that minimizes the total cost. To formulate this problem as an integer program, let

and

With these notations, the equipment scheduling problem is to

This model is often called a set-partitioning problem, since the set of legs gets divided, or partitioned, among the various routes.

The flight crews do not necessarily follow the same aircraft around a route. The main reason is that the constraints that apply to flight crews differ from those for the aircraft (for example, flight crews need to sleep occasionally). Hence, the problem has a different set of potential routes. Also, it is sometimes reasonable to allow crews to ride as passengers on some legs with the aim of getting in position for a subsequent flight. With these changes, the crew scheduling problem is

This model is often referred to as a set-covering problem, since the crews are assigned so as to cover each leg.

2 The Traveling Salesman Problem

Consider a salesman who needs to visit each of n cities, which we shall enumerate as 0, 1, …, n − 1. His goal is to start from his home city, 0, and make a tour visiting each of the remaining cities once and only once and then returning to his home. We assume that the “distance” between each pair of cities, c ij , is known (distance does not necessarily have to be distance—it could be travel time or, even better, the cost of travel) and that the salesman wants to make the tour that minimizes the total distance. This problem is called the traveling salesman problem. Figure 23.1 shows an example with seven cities. Clearly, a tour is determined by listing the cities in the order in which they will be visited. If we let s i denote the ith city visited, then the tour can be described simply as

The total number of possible tours is equal to the number of ways one can permute the n − 1 cities, i.e., (n − 1)! . Factorials are huge even for small n (for example, 50!  = 3. 041 ×10⁶⁴). Hence, enumeration is out of the question. Our aim is to formulate this problem as an integer program that can be solved more quickly than by using enumeration. It seems reasonable to introduce for each (i, j) a decision variable x ij that will be equal to one if the tour visits city j immediately after visiting city i; otherwise, it will be equal to zero. In terms of these variables, the objective function is easy to write:

The tricky part is to formulate constraints to guarantee that the set of nonzero x ij ’s corresponds exactly to a bonafide tour. Some of the constraints are fairly obvious. For example, after the salesman visits city i, he must go to one and only one city next. We can write these constraints as

(23.1)

(we call them the go-to constraints). Similarly, when the salesman visits a city, he must have come from one and only one prior city. That is,

(23.2)

(by analogy we call these the come-from constraints). If the go-to and the come-from constraints are sufficient to ensure that the decision variables represent a tour, the traveling salesman problem would be quite easy to solve because it would just be an assignment problem, which can be solved efficiently by the simplex method. But unfortunately, these constraints are not sufficient, since they do not rule out the possibility of forming disjoint subtours. An example is shown in Figure 23.2. We need to introduce more constraints to guarantee connectivity of the graph that the tour represents. To see how to do this, consider a specific tour

Let t i for i = 0, 1, …, n be defined as the number of the stop along the tour at which city i is visited; i.e., “when” city i is visited along the tour. From this definition, we see that $$t_{0} = 0,\;t_{s_{1}} = 1,\;t_{s_{2}} = 2$$, etc. In general,

so that we can think of the t j ’s as being the inverse of the s i ’s. For a bonafide tour,

Also, each t i is an integer between 0 and n − 1, inclusive. Hence, t j satisfies the following constraints:

(Note that by subtracting n in the x ij  = 0 case, we have effectively made the condition always hold.) These constraints can be written succinctly as

(23.3)

Now, these constraints were derived based on conditions that a bonafide tour satisfies. It turns out that they also force a solution to be a bonafide tour. That is, they rule out subtours. To see this, suppose to the contrary that there exists a solution to (23.1), (23.2), and (23.3) that consists of at least two subtours. Consider a subtour that does not include city 0. Let r denote the number of legs on this subtour. Clearly, r ≥ 2. Now, sum (23.3) over all arcs on this subtour. On the left, we get the sum of the t j ’s over each city visited by the subtour. On the right, we get the same sum plus r. Cancelling the sums from the two sides, we get that

which is a contradiction. Hence, the traveling salesman problem can be formulated as the following integer programming problem:

Note that, for the n-city problem, there are n ² + n variables in this formulation.

FIGURE 23.1

A feasible tour in a seven-city traveling salesman problem.

FIGURE 23.2

Two disjoint subtours in a seven-city traveling salesman problem.

3 Fixed Costs

The terms in an objective function often represent costs associated with engaging in an activity. Until now, we’ve always assumed that each of these terms is a linear function such as cx. However, it is sometimes more realistic to assume that there is a fixed cost for engaging in the activity plus a linear variable cost. That is, one such term might have the form

If we assume that there is an upper bound on the size of x, then it turns out that such a function can be equivalently modeled using strictly linear functions at the expense of introducing one integer-valued variable. Indeed, suppose that u is an upper bound on the x variable. Let y denote a {0, 1}-valued variable that is one when and only when x > 0. Then

Also, the condition that y is one exactly when x > 0 can be guaranteed by introducing the following constraints:

Of course, if the objective function has several terms with associated fixed costs, then this trick must be used on each of these terms.

4 Nonlinear Objective Functions

Sometimes the terms in the objective function are not linear at all. For example, one such term could look like the function shown in Figure 23.3. In Chapter 25, we will discuss efficient algorithms that can be used in the presence of nonlinear objective functions—at least when they have appropriate convexity/concavity properties. In this section, we will show how to formulate an integer programming approximation to a general nonlinear term in the objective function. The first step is to approximate the nonlinear function by a continuous piecewise linear function. The second step is to introduce integer variables that allow us to represent the piecewise linear function using linear relations. To see how to do this, first we decompose the variable x into a sum,

where x i denotes how much of the interval [0, x] is contained in the ith linear segment of the piecewise linear function (see Figure 23.4). Of course, some of the initial segments will lie entirely within the interval [0, x], one segment will lie partially in and partially out, and then the subsequent segments will lie entirely outside of the interval. Hence, we need to introduce constraints to guarantee that the initial x i ’s are equal to the length of their respective segments and that after the straddling segment the subsequent x i ’s are all zero. A little thought reveals that the following constraints do the trick:

Indeed, it follows from these constraints that w j  ≤ w j − 1 for j = 1, 2, …, k. This inequality implies that once one of the w j ’s is zero, then all the subsequent ones must be zero. If $$w_{j} = w_{j-1} = 1$$, the two-sided inequality on x j reduces to $$L_{j} \leq x_{j} \leq L_{j}$$. That is, x j  = L j . Similarly, if $$w_{j} = w_{j-1} = 0$$, then the two-sided inequality reduces to x j  = 0. The only other case is when w j  = 0 but $$w_{j-1} = 1$$. In this case, the two-sided inequality becomes 0 ≤ x j  ≤ L j . Therefore, in all cases, we get what we want. Now with this decomposition we can write the piecewise linear function as

FIGURE 23.3

A nonlinear function and a piecewise linear approximation to it.

FIGURE 23.4

A piecewise linear function.

5 Branch-and-Bound

In the previous sections, we presented a variety of problems that can be formulated as integer programming problems. As it happens, all of them had the property that the integer variables took just one of two values, namely, zero or one. However, there are other integer programming problems in which the integer variables can be any nonnegative integer. Hence, we define the standard integer programming problem as follows:

In this section, we shall present an algorithm for solving these problems. The algorithm is called branch-and-bound. It involves solving a (potentially) large number of (related) linear programming problems in its search for an optimal integer solution. The algorithm starts out with the following wishful approach: first ignore the constraint that the components of x be integers, solve the resulting linear programming problem, and hope that the solution vector has all integer components. Of course, hopes are almost always unfulfilled, and so a backup strategy is needed. The simplest strategy would be to round each resulting solution value to its nearest integer value. Unfortunately, this naive strategy can be quite bad. In fact, the integer solution so obtained might not even be feasible, which shouldn’t be surprising, since we know that the solution to a linear programming problem is at a vertex of the feasible set and so it is quite expected that naive movement will go outside of the feasible set.

To be concrete, consider the following example:

The linear programming problem obtained by dropping the integrality constraint is called the LP-relaxation. Since it has fewer constraints, its optimal solution provides an upper bound ζ ⁰ on the the optimal solution ζ  ∗  to the integer programming problem. Figure 23.5 shows the feasible points for the integer programming problem as well as the feasible polytope for its LP-relaxation. The solution to the LP-relaxation is at $$(x_{1},x_{2}) = (5/3,10/3)$$, and the optimal objective value is $$205/3 = 68.33$$. Rounding each component of this solution to the nearest integer, we get (2, 3), which is not even feasible. The feasible integer solution that is closest to the LP-optimal solution is (1, 3), but we can see from Figure 23.5 that this solution is not the optimal solution to the integer programming problem. In fact, it is easy to see from the figure that the optimal integer solution is either (1, 4) or (4, 0). To make the problem interesting, we’ve chosen the objective function to make the more distant point (4, 0) be the optimal solution. Of course, we can solve only very small problems by the graphical method: to solve larger problems, an algorithm is required, which we now describe. Consider variable x ₁ in the optimal solution to the LP-relaxation. Its value is 5 ∕ 3. In the optimal solution to the integer programming problem, it will be an integer. Hence, it will satisfy either x ₁ ≤ 1 or x ₁ ≥ 2. We consider these two cases separately. Let P ₁ denote the linear programming problem obtained by adding the constraint x ₁ ≤ 1 to the LP-relaxation, and let P ₂ denote the problem obtained by including the other possibility, x ₁ ≥ 2. The feasible regions for P ₁ and P ₂ are shown in Figure 23.6. Let us study P ₁ first. It is clear from Figure 23.6 that the optimal solution is at (x ₁, x ₂) = (1, 4) with an objective value of 65. Our algorithm has found its first feasible solution to the integer programming problem. We record this solution as the best-so-far. Of course, better ones may (in this case, will) come along later. Now let’s consider P ₂. Looking at Figure 23.6 and doing a small amount of calculation, we see that the optimal solution is at $$(x_{1},x_{2}) = (2,20/7)$$. In this case, the objective function value is $$478/7 = 68.29$$. Now if this value had turned out to be less than the best-so-far value, then we’d be done, since any integer solution that lies within the feasible region for P ₂ would have a smaller value yet. But this is not the case, and so we must continue our systematic search. Since $$x_{2} = 20/7 = 2.86$$, we divide P ₂ into two subproblems, one in which the constraint x ₂ ≤ 2 is added and one with x ₂ ≥ 3 added.

FIGURE 23.5

An integer programming problem. The dots represent the feasible integer points, and the shaded region shows the feasible region for the LP-relaxation.

FIGURE 23.6

The feasible subregions formed by the first branch.

Before considering these two new cases, note that we are starting to develop a tree of linear programming subproblems. This tree is called the enumeration tree. The tree as far as we have investigated is shown in Figure 23.7. The double box around P ₁ indicates that that part of the tree is done: i.e., there are no branches emanating from P ₁—it is a leaf node. The two empty boxes below P ₂ indicate two subproblems that have yet to be studied. Let’s proceed by looking at the left branch, which corresponds to adding the constraint x ₂ ≤ 2 to what we had before. We denote this subproblem by P ₃. Its feasible region is shown in Figure 23.8, from which we see that the optimal solution is at (2. 6, 2). The associated optimal objective value is 68. 2. Again, the solution is fractional. Hence, the process of subdividing must continue. This time we subdivide based on the values of x ₁. Indeed, we consider two cases: either x ₁ ≤ 2 or x ₁ ≥ 3.

FIGURE 23.7

The beginnings of the enumeration tree.

FIGURE 23.8

The refinement of P ₂ to P ₃.

Figure 23.9 shows the enumeration tree as it now stands. At this juncture, there are three directions in which we could proceed. We could either study the other branch under P ₂ or work on one of the two branches sitting under P ₃. If we were to systematically solve all the problems on a given level of the tree before going deeper, we would be performing what is referred to as a breadth-first search. On the other hand, going deep before going wide is called a depth-first search. For reasons that we shall explain later, it turns out to be better to do a depth-first search. And, to be specific, let us always choose the left branch before the right branch (in practice, there are much better rules that one can employ here). So our next linear programming problem is the one that we get by adding the constraint that x ₁ ≤ 2 to the constraints that defined P ₃. Let us call this new problem P ₄. Its feasible region is shown in Figure 23.10. It is easy to see that the optimal solution to this problem is (2, 2), with an objective value of 58. This solution is an integer solution, so it is feasible for the integer programming problem. But it is not better than our best-so-far. Nonetheless, we do not need to consider any further subproblems below this one in the enumeration tree.

FIGURE 23.9

The enumeration tree after solving P ₃.

FIGURE 23.10

The refinement of P ₃ to P ₄.

Since problem P ₄ is a leaf in the enumeration tree, we need to work back up the tree looking for the first node that has an unsolved problem sitting under it. For the case at hand, the unsolved problem is on the right branch underneath P ₃. Let us call this problem P ₅. It too is depicted in Figure 23.10. The optimal solution is (3, 1. 43), with an optimal objective function value of 68. 14. Since this objective function value is larger than the value of the best-so-far integer solution, we must further investigate by dividing into two possibilities, either x ₂ ≤ 1 or x ₂ ≥ 2. At this point, the enumeration tree looks like that shown in Figure 23.11.

FIGURE 23.11

The enumeration tree after solving P ₅. The double box around P ₄ indicates that it is a leaf in the tree.

Let P ₆ denote the linear programming problem that we get on the left branch under P ₅. Its feasible region is shown in Figure 23.12. The optimal solution is (3. 3, 1), and the associated objective value is 68. 1. Again, the solution is fractional and has a higher objective value than the best-so-far integer solution. Hence, it must be subdivided based on x ₁ ≤ 3 as opposed to x ₁ ≥ 4. Denoting these two problems by P ₇ and P ₈, their feasible regions are as depicted in Figure 23.13. The solution to P ₇ is (3, 1), and the objective value is 63. This is an integer solution, but it is not better than the best-so-far. Nonetheless, the node becomes a leaf, since the solution is integral. Hence, we move on to P ₈. The solution to this problem is also integral, (4, 0). Also, the objective value associated with this solution is 68, which is a new record for feasible integer solutions. Hence, this solution becomes our best-so-far. The enumeration tree at this point is shown in Figure 23.14.

FIGURE 23.12

The refinement of P ₅ to P ₆.

FIGURE 23.13

The refinement of P ₆ to P ₇ and P ₈.

FIGURE 23.14

The enumeration tree after solving P ₆, P ₇, and P ₈.

Now we need to go back and solve the problems under P ₅ and P ₂ (and any subproblems thereof). It turns out that both these subproblems are infeasible, and so no more subdivisions are needed. The enumeration tree is now completely fathomed and is shown in Figure 23.15. We can now assert that the optimal solution to the original integer programming problem was found in problem P ₈. The solution is (x ₁, x ₂) = (4, 0), and the associated objective function value is 68.

There are three reasons why depth-first search is generally the preferred order in which to fathom the enumeration tree. The first is based on the observation that most integer solutions lie deep in the tree. There are two advantages to finding integer feasible solutions early. The first is simply the fact that it is better to have a feasible solution than nothing in case one wishes to abort the solution process early. But more importantly, identifying an feasible integer solution can result in subsequent nodes of the enumeration tree being made into leaves simply because the optimal objective function associated with that node is lower than the best-so-far integer solution. Making such nodes into leaves is called pruning the tree and can account for tremendous gains in efficiency.

FIGURE 23.15

The complete enumeration tree.

A second reason to favor depth-first search is the simple fact that it is very easy to code the algorithm as a recursively defined function. This may seem trite, but one shouldn’t underestimate the value of code simplicity when implementing algorithms that are otherwise quite sophisticated, such as the one we are currently describing.

The third reason to favor depth-first search is perhaps the most important. It is based on the observation that as one moves deeper in the enumeration tree, each subsequent linear programming problem is obtained from the preceding one by simply adding (or refining) an upper/lower bound on one specific variable. To see why this is an advantage, consider for example problem P ₂, which is a refinement of P ₀. The optimal dictionary for problem P ₀ is recorded as

Problem P ₂ is obtained from P ₀ by adding the constraint that x ₁ ≥ 2. Introducing a variable, g ₁, to stand for the difference between x ₁ and this lower bound and using the dictionary above to write x ₁ in terms of the nonbasic variables, we get

Therefore, we can use the following dictionary as a starting point for the solution of P ₂:

This dictionary is dual feasible but primal infeasible. Therefore, the dual simplex method is likely to find a new optimal solution in very few iterations. According to the dual simplex method, variable g ₁ is the leaving variable and w ₂ is the corresponding entering variable. Making the pivot, we get the following dictionary:

This dictionary is optimal for P ₂. In general, the dual simplex method will take more than one iteration to reoptimize, but nonetheless, one does expect it to get to a new optimal solution quickly.

We end this chapter by remarking that many real problems have the property that some variables must be integers but others can be real valued. Such problems are called mixed integer programming problems. It should be easy to see how to modify the branch-and-bound technique to handle such problems as well.

Exercises

23.1

Knapsack Problem. Consider a picnicker who will be carrying a knapsack that holds a maximum amount b of “stuff.” Suppose that our picnicker must decide what to take and what to leave behind. The jth thing that might be taken occupies a j units of space in the knapsack and will bring c j amount of “enjoyment.” The knapsack problem then is to maximize enjoyment subject to the constraint that the stuff brought must fit into the knapsack:

This apparently simple problem has proved difficult for general-purpose branch-and-bound algorithms. To see why, analyze the special case in which each thing contributes the same amount of enjoyment, i.e., c j  = c for all j, and takes up exactly two units of space, i.e., a j  = 2 for all j. Suppose also that the knapsack holds n units of stuff.

(a)

What is the optimal solution when n is even? when n is odd?

(b)

How many subproblems must the branch-and-bound algorithm consider when n is odd?

23.2

Vehicle Routing. Consider the dispatching of delivery vehicles (for example, mail trucks, fuel-oil trucks, newspaper delivery trucks, etc.). Typically, there is a fleet of vehicles that must be routed to deliver goods from a depot to a given set of n drop-points. Given a set of feasible delivery routes and the cost associated with each one, explain how to formulate the problem of minimizing the total delivery cost as a set-partitioning problem.

23.3

Explain how to modify the integer programming reformulation of continuous piecewise linear functions so that it covers piecewise linear functions having discontinuities at the junctions of the linear segments. Can fixed costs be handled with this approach?

Notes

Standard references for integer programming include the classic text by Garfinkel and Nemhauser (1972) and the more recent text by Nemhauser and Wolsey (1988). Bill Cook’s recent book Cook (2012) on the traveling salesman problem gives an entertaining historical and mathematical perspective on this particular application of integer programming.

Bibliography

Cook, W. (2012). In pursuit of the traveling salesman. Princeton: Princeton University Press.

Garfinkel, R., and Nemhauser, G. (1972). Integer programming. New York: Wiley.

Nemhauser, G., and Wolsey, L. (1988). Integer and combinatorial optimization. New York: Wiley.

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_24

24. Quadratic Programming

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In Chapter 23, we studied a generalization of the linear programming problem in which variables were constrained to take on integer values. In this chapter, we consider a generalization of a different kind. Namely, we shall study the class of problems that would be linear programs except that the objective function is permitted to include terms involving products of pairs of variables. Such terms are called quadratic terms, and the problems we shall study are called quadratic programming problems.

We have two reasons for being interested in quadratic programming problems. First, on the practical side, there are many real-world optimization problems that fall into this category. This is so because most real-world applications have an element of uncertainty to them and that uncertainty is modeled by including a sum of squares deviation, i.e. variance, as a measure of the robustness of the solution. It is often possible to arrange it so that these quadratic robustness terms appear only in the objective function. The quadratic version of the portfolio selection problem studied in Chapter 13 is one such example—there are many others. The second reason for our interest in quadratic programming problems is that they form a bridge to the much broader subject of convex programming that we shall take up in Chapter 25.

We begin this chapter with a quadratic variant of the portfolio selection problem.

1 The Markowitz Model

Harry Markowitz received the 1990 Nobel Prize in Economics for his portfolio optimization model in which the tradeoff between risk and reward is explicitly treated. We shall briefly describe this model in its simplest form. We start by reintroducing the basic framework of the problem. Those who have read Chapter 13 will note that the first few paragraphs here are a repeat of what was written there.

Given a collection of potential investments (indexed, say, from 1 to n), let R j denote the return in the next time period on investment j, j = 1, …, n. In general, R j is a random variable, although some investments may be essentially deterministic.

A portfolio is determined by specifying what fraction of one’s assets to put into each investment. That is, a portfolio is a collection of nonnegative numbers x j , j = 1, …, n, that sum to one. The return (on each dollar) one would obtain using a given portfolio is given by

The reward associated with such a portfolio is defined as the expected return:

If reward alone were the issue, the problem would be trivial: simply put everything in the investment with the highest expected return. But unfortunately, investments with high reward typically also carry a high level of risk. That is, even though they are expected to do very well in the long run, they also tend to be erratic in the short term. Markowitz defined the risk associated with an investment to be the variance of the return:

where $$\tilde{R}_{j} = R_{j} - \mathbb{E}R_{j}$$. One would like to maximize the reward while at the same time not incur excessive risk. In the Markowitz model, one forms a linear combination of the mean and the variance (parametrized here by μ) and minimizes that:

(24.1)

Here, as in Chapter 13, μ is a positive parameter that represents the importance of risk relative to reward: high values of μ tend to minimize risk at the expense of reward, whereas low values put more weight on reward.

Again, as in Chapter 13, whenever there are individual investments that are negatively correlated, i.e. one is likely to go up exactly on those days where the other is likely to go down, it is wise to buy some of each. This is called hedging. In statistics, the so-called covariance matrix is the key to identifying negative correlations. And, the covariance matrix is what appears in the Markowitz model. To see it, let us expand the square in our expression for the variance of the portfolio:

where

is the covariance matrix. Hence, problem (24.1) can be rewritten as

(24.2)

where we have introduced $$r_{j} = \mathbb{E}R_{j}$$ for the mean return on investment j.

Solving problem (24.2) requires an estimate of the mean return for each of the investments as well as an estimate of the covariance matrix. However, these quantities are not known theoretically but instead must be estimated by looking at historical data. For example, Table 24.1 shows annual returns from 1973 to 1994 for eight different possible investments: U.S. 3-Month T-Bills, U.S. Government Long Bonds, S&P 500, Wilshire 5000 (a collection of small company stocks), NASDAQ Composite, Lehman Brothers Corporate Bonds Index, EAFE (a securities index for Europe, Asia, and the Far East), and Gold. Let R j (t) denote the return on investment j in year 1972 + t. One way to estimate the mean $$\mathbb{E}R_{j}$$ is simply to take the average of the historical returns:

There are two drawbacks to this simple formula. First, whatever happened in 1973 certainly has less bearing on the future than what happened in 1994. Hence, giving all the past returns equal weight puts too much emphasis on the distant past at the expense of the recent past. A better estimate is obtained by using a discounted sum:

Here, p is a discount factor. Putting p = 0. 9 gives a weighted average that puts more weight on the most recent years. To see the effect of discounting the past, consider the Gold investment. The unweighted average return is 1. 129, whereas the weighted average is 1. 053. Most experts in 1995 felt that a 5. 3% return represented a more realistic expectation than a 12. 9% return. In the results that follow, all expectations are estimated by computing weighted averages using p = 0. 9.

Table 24.1

Returns per dollar for each of eight investments over several years. That is, $1 invested in US 3-Month T-Bills on January 1, 1973, was worth $1. 075 on December 31, 1973.

Year

US

US

S&P

Wilshire

NASDAQ

Lehman

EAFE

Gold

3-Month

Gov.

500

5000

Composite

Bros.

T-Bills

Long

Corp.

Bonds

Bonds

1973

1.075

0.942

0.852

0.815

0.698

1.023

0.851

1.677

1974

1.084

1.020

0.735

0.716

0.662

1.002

0.768

1.722

1975

1.061

1.056

1.371

1.385

1.318

1.123

1.354

0.760

1976

1.052

1.175

1.236

1.266

1.280

1.156

1.025

0.960

1977

1.055

1.002

0.926

0.974

1.093

1.030

1.181

1.200

1978

1.077

0.982

1.064

1.093

1.146

1.012

1.326

1.295

1979

1.109

0.978

1.184

1.256

1.307

1.023

1.048

2.212

1980

1.127

0.947

1.323

1.337

1.367

1.031

1.226

1.296

1981

1.156

1.003

0.949

0.963

0.990

1.073

0.977

0.688

1982

1.117

1.465

1.215

1.187

1.213

1.311

0.981

1.084

1983

1.092

0.985

1.224

1.235

1.217

1.080

1.237

0.872

1984

1.103

1.159

1.061

1.030

0.903

1.150

1.074

0.825

1985

1.080

1.366

1.316

1.326

1.333

1.213

1.562

1.006

1986

1.063

1.309

1.186

1.161

1.086

1.156

1.694

1.216

1987

1.061

0.925

1.052

1.023

0.959

1.023

1.246

1.244

1988

1.071

1.086

1.165

1.179

1.165

1.076

1.283

0.861

1989

1.087

1.212

1.316

1.292

1.204

1.142

1.105

0.977

1990

1.080

1.054

0.968

0.938

0.830

1.083

0.766

0.922

1991

1.057

1.193

1.304

1.342

1.594

1.161

1.121

0.958

1992

1.036

1.079

1.076

1.090

1.174

1.076

0.878

0.926

1993

1.031

1.217

1.100

1.113

1.162

1.110

1.326

1.146

1994

1.045

0.889

1.012

0.999

0.968

0.965

1.078

0.990

The second issue concerns the estimation of means (not variances). An investment that returns 1. 1 one year and 0. 9 the next has an (unweighted) average return of 1, that is, no gain or loss. However, one dollar invested will actually be worth (1. 1)(0. 9) = 0. 99 at the end of the second year. While a 1% error is fairly small, consider what happens if the return is 2. 0 one year and then 0. 5 the next. Clearly, the value of one dollar at the end of the 2 years is (2. 0)(0. 5) = 1, but the average of the two returns is $$(2.0 + 0.5)/2 = 1.25$$. There is a very significant difference between an investment that is flat and one that yields a 25% return in 2 years. This is obviously an effect for which a correction is required. We need to average 2. 0 and 0. 5 in such a way that they cancel out—and this cancellation must work not only for 2. 0 and 0. 5 but for every positive number and its reciprocal. The trick is to average the logarithm of the returns (and then exponentiate the average). The logarithm has the correct effect of cancelling a return r and its reciprocal:

Hence, we estimate means from Table 24.1 using

This estimate for Gold gives an estimate of its return at 2. 9%, which is much more in line with the beliefs of experts (at least in 1995).

Table 24.2 shows the optimal portfolios for several choices of μ. The corresponding optimal values for the mean and standard deviation (which is defined as the square root of the variance) are plotted in Figure 24.1. Letting μ vary continuously generates a curve of optimal solutions. This curve is called the efficient frontier. Any portfolio that produces a mean–variance combination that does not lie on the efficient frontier can be improved either by increasing its mean without changing the variance or by decreasing the variance without changing the mean. Hence, one should only invest in portfolios that lie on the efficient frontier.

Table 24.2

Optimal portfolios for several choices of μ.

μ

Gold

US

Lehman

NASDAQ

S&P

EAFE

Mean

Std.

3-Month

Bros.

Composite

500

dev.

T-Bills

Corp.

Bonds

0.0

1.000

1.122

0.227

0.1

0.603

0.397

1.121

0.147

1.0

0.876

0.124

1.120

0.133

2.0

0.036

0.322

0.549

0.092

1.108

0.102

4.0

0.487

0.189

0.261

0.062

1.089

0.057

8.0

0.713

0.123

0.117

0.047

1.079

0.037

1024.0

0.008

0.933

0.022

0.016

0.022

1.070

0.028

FIGURE 24.1

The efficient frontier.

Of course, the optimal portfolios shown in Table 24.2 were obtained by solving (24.1). The rest of this chapter is devoted to describing an algorithm for solving quadratic programs such as this one.

2 The Dual

We have seen that duality plays a fundamental role in our understanding and derivation of algorithms for linear programming problems. The same is true for quadratic programming. Hence, our first goal is to figure out what the dual of a quadratic programming problem should be.

Quadratic programming problems are usually formulated as minimizations. Therefore, we shall consider problems given in the following form:

(24.3)

Of course, we may (and do) assume that the matrix Q is symmetric (see Exercise 24.2). Note that we have also changed the sense of the inequality constraints from our usual less-than to greater-than. This change is not particularly important—its only purpose is to maintain a certain level of parallelism with past formulations (that is, minimizations have always gone hand-in-hand with greater-than constraints, while maximizations have been associated with less-than constraints).

In Chapter 5, we derived the dual problem by looking for tight bounds on the optimal solution to the primal problem. This approach could be followed here, but it seems less compelling in the context of quadratic programming. A more direct approach stems from the connection between duality and the first-order optimality conditions for the barrier problem that we examined in Chapter 17. Indeed, let us start by writing down the barrier problem associated with (24.3). To this end, we introduce a nonnegative vector w of surplus variables and then subtract a barrier term for each nonnegative variable to get the following barrier problem:

Next, we introduce the Lagrangian:

The first-order optimality conditions for the barrier problem are obtained by differentiating the Lagrangian with respect to each of its variables and setting these derivatives to zero. In vector notation, setting to zero the derivative with respect to the x variables gives

Similarly, setting to zero the derivatives with respect to the w and y variables gives

respectively. As we did in our study of linear programming problems, we now introduce a new vector z given by

With this definition, the first-order optimality conditions can be summarized as

From the last two conditions, we see that the dual problem involves an n-vector of variables z that are complementary to the primal variables x and an m-vector of variables y that are complementary to the primal slack variables w. Because of these complementarity conditions, we expect that the variables y and z are constrained to be nonnegative in the dual problem. Also, to establish the proper connection between the first-order optimality conditions and the dual problem, we must recognize the first condition as a dual constraint. Hence, the constraints for the dual problem are

It is interesting to note that the dual constraints involve an n-vector x that seems as if it should belong to the primal problem. This may seem odd, but when understood properly it turns out to be entirely harmless. The correct interpretation is that the variable x appearing in the dual has, in principle, no connection to the variable x appearing in the primal (except that, as we shall soon see, at optimality they will be equal).

The barrier problem has helped us write down the dual constraints, but it does not shed any light on the dual objective function. To see what the dual objective function should be, we look at what it needs to be for the weak duality theorem to hold true. In the weak duality theorem, we assume that we have a primal feasible solution (x, w) and a dual feasible solution (x, y, z). We then follow the obvious chains of equalities:

and

Now, since $${y}^{T}(Ax) = {({A}^{T}y)}^{T}x$$, we see that

From this inequality, we see that the dual objective function is $${b}^{T}y -\frac{1} {2}{x}^{T}Qx$$. Hence, the dual problem can be stated now as

For linear programming, the fundamental connection between the primal and dual problems is summarized in the Complementary Slackness Theorem. In the next section, we shall derive a version of this theorem for quadratic programming.

3 Convexity and Complexity

In linear programming, the dual problem is important because it provides a certificate of optimality as manifest in the Complementary Slackness Theorem. Under certain conditions, the same is true here. Let us start by deriving the analogue of the Complementary Slackness Theorem. The derivation begins with a reiteration of the derivation of the Weak Duality Theorem. Indeed, let (x, w) denote a feasible solution to the primal problem and let $$(\bar{x},y,z)$$ denote a feasible solution to the dual problem (we have put a bar on the dual x to distinguish it from the one appearing in the primal). The chain of equalities that form the backbone of the proof of the Weak Duality Theorem are, as always, obtained by writing y T Ax two ways, namely,

and then producing the obvious substitutions

and

Comparing the ends of these two chains and using the fact that both y T w and z T x are nonnegative, we see that

(24.4)

So far, so good.

Now, what about the Complementary Slackness Theorem? In the present context, we expect this theorem to say roughly the following: given a solution $$({x}^{{\ast}},{w}^{{\ast}})$$ that is feasible for the primal and a solution $$({x}^{{\ast}},{y}^{{\ast}},{z}^{{\ast}})$$ that is feasible for the dual, if these solutions make inequality (24.4) into an equality, then the primal solution is optimal for the primal problem and the dual solution is optimal for the dual problem.

Let’s try to prove this. Let (x, w) be an arbitrary primal feasible solution. Weak duality applied to (x, w) on the primal side and $$({x}^{{\ast}},{y}^{{\ast}},{z}^{{\ast}})$$ on the dual side says that

But for the specific primal feasible solution $$({x}^{{\ast}},{w}^{{\ast}})$$, this inequality is an equality:

Combining these, we get

This is close to what we want, but not quite it. Recall that our aim is to show that the primal objective function evaluated at x  ∗  is no larger than its value at x. That is,

It is easy to get from the one to the other. Starting from the desired left-hand side, we compute as follows:

The last step in the derivation is to drop the subtracted term on the right-hand side of the last expression. We can do this if the quantity being subtracted is nonnegative. But is it? In general, the answer is no. For example, if Q were the negative of the identity matrix, then the expression $${(x - {x}^{{\ast}})}^{T}Q(x - {x}^{{\ast}})$$ would be negative rather than nonnegative.

So it is here that we must impose a restriction on the class of quadratic programming problems that we study. The correct assumption is that Q is positive semidefinite. Recall from Chapter 19 that a matrix Q is positive semidefinite if

With this assumption, we can finish the chain of inequalities and conclude that

Since x was an arbitrary primal feasible point, it follows that x  ∗  (together with w  ∗ ) is optimal for the primal problem. A similar analysis shows that y  ∗  (together with x  ∗  and z  ∗ ) is optimal for the dual problem (see Exercise 24.4).

A quadratic programming problem of the form (24.3) in which the matrix Q is positive semidefinite is called a convex quadratic programming problem. The discussion given above can be summarized in the following theorem:

Theorem 24.1.

For convex quadratic programming problems, given a solution $$({x}^{{\ast}},{w}^{{\ast}})$$ that is feasible for the primal and a solution $$({x}^{{\ast}},{y}^{{\ast}},{z}^{{\ast}})$$ that is feasible for the dual, if these solutions make inequality (24.4)into an equality, then the primal solution is optimal for the primal problem and the dual solution is optimal for the dual problem.

To see how bad things are when Q is not positive semidefinite, consider the following example:

(24.5)

We assume that the coefficients, c j , j = 1, 2, …, n, are small. To be precise, we assume that

Let f(x) denote the value of the objective function at point x. Setting the gradient to zero,

we see that there is one interior critical point. It is given by

(the assumption that c is small guarantees that this x lies in the interior of the feasible set: 0 < x < 1). However, this critical point is a local maximum, since the matrix of second derivatives is − 2I. The algebraic details are tedious, but if we look at Figure 24.2, it is easy to be convinced that every vertex of the feasible set is a local minimum. While this particular problem is easy to solve explicitly, it does indicate the essential difficulty associated with nonconvex quadratic programming problems—namely, for such problems one may need to check every vertex individually, and there may be an exponential number of such vertices.

FIGURE 24.2

The objective function for (24.5) in the case where n = 2.

The situation for convex quadratic programming problems is much better, since they inherit most of the properties that made linear programs efficiently solvable. Indeed, in the next section, we derive an interior-point method for quadratic programming problems.

4 Solution via Interior-Point Methods

In this section, we derive an interior-point method for quadratic programming problems. We start from the first-order optimality conditions, which we saw in the last section are given by

Following the derivation given in Chapter 18, we replace (x, w, y, z) with $$(x + \Delta x,w + \Delta w,y + \Delta y,z + \Delta z)$$ to get the following nonlinear system in (Δx, Δw, Δy, Δz):

Next, we drop the nonlinear terms to get the following linear system for the step directions (Δx, Δw, Δy, Δz):

Following the reductions of Chapter 19, we use the last two equations to solve for Δz and Δw to get

We then use these expressions to eliminate Δz and Δw from the remaining two equations in the system. After elimination, we arrive at the following reduced KKT system:

(24.6)

(24.7)

Substituting in the definitions of ρ and σ and writing the system in matrix notation, we get

A summary of the algorithm is shown in Figure 24.3.

FIGURE 24.3

The path-following method for quadratic programming problems.

It should be clear that the quadratic term in the objective function plays a fairly small role. In fact, the convergence analysis given in Chapter 18 can be easily adapted to yield analogous results for quadratic programming problems (see Exercise 24.6).

5 Practical Considerations

For practical implementations of interior-point algorithms, we saw in Chapter 19 that the difficulties created by dense rows/columns suggest that we solve the reduced KKT system using an equation solver that can handle symmetric indefinite systems (such as those described in Chapter 20). Quadratic programming problems give us even more reason to prefer the reduced KKT system. To see why, let us reduce the system further to get a feel for the normal equations for quadratic programming.

If we use (24.6) to solve for Δx and then eliminate it from (24.7), we get

and the associated system of normal equations (in primal form):

As we saw in Chapter 19, the most significant disadvantage of the normal equations is that they could involve a dense matrix even when the original constraint matrix is sparse. For quadratic programming, this disadvantage is even more pronounced. Now the matrix of normal equations has the nonzero pattern of $$A{(D + Q)}^{-1}{A}^{T}$$, where D is a diagonal matrix. If Q is a diagonal matrix, then this matrix appearing between A and A T is diagonal, and the system has the same structure as we saw for linear programming. But if Q is not a diagonal matrix, then all hope for any sparsity in $$A{(D + Q)}^{-1}{A}^{T}$$ is lost.

Fortunately, however, the dual form of the normal equations is likely to retain some sparsity. Indeed, to derive the dual form, we use (24.7) to solve for Δy and then eliminate it from (24.6). The result is

and

Now the matrix has a nonzero pattern of A T A + Q. This pattern is much more likely to be sparse than the pattern we had above.

As mentioned earlier, there is significantly less risk of fill-in if Q is diagonal. A quadratic programming problem for which Q is diagonal is called a separable quadratic programming problem. It turns out that every nonseparable quadratic programming problem can be replaced by an equivalent separable version, and sometimes this replacement results in a problem that can be solved dramatically faster than the original nonseparable problem. The trick reveals itself when we remind ourselves that the problems we are studying are convex quadratic programs, and so we ask the question: how do we know that the matrix Q is positive semidefinite? Or, more to the point, how does the creator of the model know that Q is positive semidefinite? There are many equivalent characterizations of positive semidefiniteness, but the one that is easiest to check is the one that says that Q is positive semidefinite if and only if it can be factored as follows:

Here F is a k ×n matrix and D is a k ×k diagonal matrix having all nonnegative diagonal entries. In fact, the model creator often started with F and D and then formed Q by multiplying. In these cases, the matrix F will generally be less dense than Q. And if k is substantially less than n, then the following substitution is almost guaranteed to dramatically improve the solution time. Introduce new variables y by setting

With this definition, the nonseparable quadratic programming problem (24.3) can be replaced by the following equivalent separable one:

The cost of separation is the addition of k new constraints. As we said before, if k is small and/or F is sparse, then we can expect this formulation to be solved more efficiently.

To illustrate this trick, let us return to the Markowitz model. Recall that the quadratic terms in this model come from the variance of the portfolio’s return, which is given by

Here,

for t = 1, 2, …, T, and

If we introduce the variables,

then we get the following separable version of the Markowitz model:

Using specific data involving 500 possible investments and 20 historical time periods, the separable version solves 60 times faster than the nonseparable version using a QP-solver called LOQO.

Exercises

24.1

Show that the gradient of the function

is given by

24.2

Suppose that Q is an n ×n matrix that is not necessarily symmetric. Let $$\tilde{Q} = \frac{1} {2}(Q + {Q}^{T})$$. Show that

(a)

$${x}^{T}Qx = {x}^{T}\tilde{Q}x$$, for every x ∈ ℝ n , and

(b)

$$\tilde{Q}$$ is symmetric.

24.3

Penalty Methods.

(a)

Consider the following problem:

where Q is symmetric, positive semidefinite, and invertible (these last two conditions are equivalent to saying that Q is positive definite). By solving the first-order optimality conditions, give an explicit formula for the solution to this problem.

(b)

Each equality constraint in the above problem can be replaced by a penalty term added to the objective function. Penalty terms should be small when the associated constraint is satisfied and become rapidly larger as it becomes more and more violated. One choice of penalty function is the quadratic function. The quadratic penalty problem is defined as follows:

where λ is a large real-valued parameter. Derive an explicit formula for the solution to this problem.

(c)

Show that, in the limit as λ tends to infinity, the solution to the quadratic penalty problem converges to the solution to the original problem.

24.4

Consider a convex quadratic programming problem. Suppose that $$({x}^{{\ast}},{w}^{{\ast}})$$ is a feasible solution for the primal and that $$({x}^{{\ast}},{y}^{{\ast}},{z}^{{\ast}})$$ is a feasible solution for the dual. Suppose further that these solutions make inequality (24.4) into an equality. Show that the dual solution is optimal for the dual problem.

24.5

A real-valued function f defined on ℝ n is called convex if, for every x, y ∈ ℝ n , and for every 0 < t < 1,

Show that the function

is convex if Q is positive semidefinite.

24.6

Extend the convergence analysis given in Chapter 18 so that it applies to convex quadratic programming problems, and identify in particular any steps that depend on Q being positive semidefinite.

24.7

Consider the quadratic programming problem given in the following form:

(i.e., without assuming nonnegativity of the x vector). Show that the formulas for the step directions Δx and Δy are given by the following reduced KKT system:

(24.8)

Notes

The portfolio optimization model presented in Section 24.1 was first introduced by Markowitz ( 1959 ). He received the 1990 Nobel Prize in Economics for this work.

Quadratic programming is the simplest class of problems from the subject called nonlinear programming. Two excellent recent texts that cover nonlinear programming are those by Bertsekas ( 1995 ) and Nash and Sofer ( 1996 ). The first paper that extended the path-following method to quadratic programming was Monteiro and Adler ( 1989 ). The presentation given here follows Vanderbei ( 1999 ).

Bibliography

Bertsekas, D. (1995). Nonlinear programming. Belmont: Athena Scientific.

Markowitz, H. (1959). Portfolio selection: Efficient diversification of investments. New York: Wiley.

Monteiro, R., and Adler, I. (1989). Interior path following primal-dual algorithms: Part i: Linear programming. Mathematical Programming, 44, 27–41.

Nash, S., and Sofer, A. (1996). Linear and nonlinear programming. New York: McGraw-Hill.

Vanderbei, R. (1999). LOQO: An interior point code for quadratic programming. Optimization Methods and Software, 12, 451–484.Crossref

© Springer Science+Business Media New York 2014

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_25

25. Convex Programming

Robert J. Vanderbei1 

(1)

Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey, USA

In the last chapter, we saw that small modifications to the primal–dual interior-point algorithm allow it to be applied to quadratic programming problems as long as the quadratic objective function is convex. In this chapter, we shall go further and allow the objective function to be a general (smooth) convex function. In addition, we shall allow the feasible region to be any convex set given by a finite collection of convex inequalities.

1 Differentiable Functions and Taylor Approximations

In this chapter, all nonlinear functions will be assumed to be twice differentiable, and the second derivatives will be assumed continuous. We begin by reiterating a few definitions and results that were briefly touched on in Chapter 17. First of all, given a real-valued function f defined on a domain in $${\mathbb{R}}^{n}$$, the vector

is called the gradient of f at x. The matrix

is called the Hessian of f at x. In dimensions greater than one, the gradient and the Hessian are the analogues of the first and second derivatives of a function in one dimension. In particular, they appear in the three-term Taylor series expansion of f about the point x:

The last term is called the remainder term. The value of this expansion lies in the fact that this remainder is small when $$\Delta x$$ is small. To be precise, the remainder has the following property:

This result follows immediately from the one-dimensional three-term Taylor series expansion applied to $$g(t) = f(x + t\Delta x)$$ and the chain rule (see Exercise 25.8).

2 Convex and Concave Functions

There are several equivalent definitions of convexity of a function. The definition that is most expedient for our purposes is the multidimensional generalization of the statement that a function is convex if its second derivative is nonnegative. Hence, we say that a real-valued function defined on a domain in $${\mathbb{R}}^{n}$$ is convex if its Hessian is positive semidefinite everywhere in its domain. A function is called concave if its negation is convex.

3 Problem Formulation

We shall study convex optimization problems posed in the following form:

Here, the real-valued function c( ⋅) is assumed to be convex, and the m real-valued functions a i ( ⋅) are assumed to be concave. This formulation is the natural extension of the convex quadratic programming problem studied in the previous chapter, except that we have omitted the nonnegativity constraints on the variables. This omission is only a matter of convenience since, if a given problem involves nonnegative variables, the assertion of their nonnegativity can be incorporated as part of the m nonlinear inequality constraints. Also note that once we allow for general concave inequality constraints, we can take the right-hand sides to be zero by simply incorporating appropriate shifts into the nonlinear constraint functions. Hence, many texts on convex optimization prefer to formulate the constraints in the form a i (x) ≥ 0. We have left the constants b i on the right-hand side for later comparisons with the quadratic programming problem of the previous chapter. Finally, note that many convex and concave functions become infinite in places and therefore have a natural domain that is a strict subset of $${\mathbb{R}}^{n}$$. This issue is important to address when solving practical problems, but since this chapter is just an introduction to convex optimization, we shall assume that all functions are finite on all of $${\mathbb{R}}^{n}$$.

At times it will be convenient to use vector notation to consolidate the m constraints into a single inequality. Hence, we sometimes express the problem as

where A( ⋅) is a function from $${\mathbb{R}}^{n}$$ into $${\mathbb{R}}^{m}$$ and b is a vector in $${\mathbb{R}}^{m}$$. As usual, we let w denote the slack variables that convert the inequality constraints to equalities:

4 Solution via Interior-Point Methods

In this section, we derive an interior-point method for convex programming problems. We start by introducing the associated barrier problem:

The Lagrangian for this problem is given by

Equating to zero the derivative of L with respect to each of its variables, we get the following set of first-order optimality conditions:

The next step is to multiply the ith equation in the middle set by w i and then replace x with $$x + \Delta x$$, y by $$y + \Delta y$$, and w by $$w + \Delta w$$ to get the following system:

Now we view this set of equations as a nonlinear system in the “delta” variables and linearize it by replacing each nonlinear function with its two-term Taylor series approximation. For example, $$\partial c/\partial x_{j}(x + \Delta x)$$ gets replaced with

Similarly, $$\partial a_{i}/\partial x_{j}(x + \Delta x)$$ gets replaced with

Writing the resulting linear system with the delta-variable terms on the left and everything else on the right, we get

(Note that we have omitted the indication that the functions c, a i , and their derivatives are to be evaluated at x.)

As usual, the next step is to solve the middle set of equations for the $$\Delta w_{i}$$’s and then to eliminate them from the system. The reduced system then becomes

and the equations for the $$\Delta w_{i}$$’s are

At this point it is convenient to put the equations into matrix form. If we generalize our familiar gradient notation by letting ∇ A(x) denote the m ×n matrix whose (i, j)th entry is $$\partial a_{i}/\partial x_{j}(x)$$, then we can write the above system succinctly as follows:

(25.1)

Now that we have step directions, the algorithm is easy to describe—just compute step lengths that preserve strict positivity of the w i ’s and the y i ’s, step to a new point, and iterate.

5 Successive Quadratic Approximations

It is instructive to notice the similarity between the system given above and the analogous system for the quadratic programming problem posed in the analogous form (see Exercise 24.7). Indeed, a careful matching of terms reveals that the step directions derived here are exactly those that would be obtained if one were to form a certain quadratic approximation at the beginning of each iteration of the interior-point algorithm. Hence, the interior-point method can be thought of as a successive quadratic programming algorithm. In order to write this quadratic approximation neatly, let $$\bar{x}$$ and $$\bar{y}$$ denote the current primal and dual variables, respectively. Then the quadratic approximation can be written as

To verify the equivalence, we first observe that this problem is a quadratic program whose linear objective coefficients are given by

whose quadratic objective coefficients are given by

and whose right-hand side vector is given by

Substituting these expressions into the appropriate places in (24.​8), we get (25.1).

Looking at the quadratic terms in the objective of the quadratic programming approximation, we see that the objective is convex, since we assumed at that start that c is convex, each a i is concave, and the dual variables multiplying the Hessians of the constraint functions are all strictly positive.

6 Merit Functions

It is perhaps a little late to bring this up, but here’s a small piece of advice: always test your knowledge on the simplest possible example. With that in mind, consider the following trivial convex optimization problem:

This problem has no constraints. Looking at the graph of the objective function, which looks like a smoothed out version of | x | , we see that the optimal solution is x  ∗  = 0. What could be easier! There are no y i ’s nor any w i ’s and equation (25.1) becomes just

where $$c(x) = \sqrt{1 + {x}^{2}}$$. Taking the first and second derivatives, we get

Substituting these expressions into the equation for $$\Delta x$$ and simplifying, we get that

Since there are no nonnegative variables that need to be kept positive, we can take unshortened steps. Hence, letting x (k) denote our current point and x (k + 1) denote the next point, we have that

That is, the algorithm says to start at any point x ⁽⁰⁾ and then replace this point with the negative of its cube, replace that with the negative of its cube, and so on.

The question is: does this sequence converge to zero? It is easy to see that the answer is yes if | x ⁽⁰⁾ | < 1 but no otherwise. For example, if we start with $${x}^{(0)} = 1/2$$, then the sequence of iterates is If, on the other hand, we start at x ⁽⁰⁾ = 2, then we get the following wildly divergent sequence: Here is what goes wrong in this example. For problems without constraints, our algorithm has an especially simple description:

From the current point, use the first three terms of a Taylor series expansion to make a quadratic approximation to the objective function. The next point is the minimum of this quadratic approximation function.

k

x (k)

0

0.50000000

1

 − 0.12500000

2

0.00195313

3

 − 0.00000001

k

x (k)

0

2

1

 − 8

2

512

3

 − 134,217,728

Figure 25.1 shows a graph of the objective function together with the quadratic approximation at x ⁽⁰⁾ = 2. It is easy to see that the next iterate is at − 8. Also, the further from zero that one starts, the more the function looks like a straight line and hence the further the minimum will be to the other side. How do we remedy this nonconvergence? The key insight is the observation that the steps are always in the correct direction (i.e, a descent direction) but they are too long—we need to shorten them. A standard technique for shortening steps in situations like this is to introduce a function called a merit function and to shorten steps as needed to ensure that this merit function is always monotonically decreasing. For the example above, and in fact for any unconstrained optimization problem, we can use the objective function itself as the merit function. But, for problems with constraints, one needs to use something a little different from just the objective function. For example, one can use the logarithmic barrier function plus a constant times the square of the Euclidean norm of the infeasibility vector:

Here, β is a positive real number. One can show that for β sufficiently large the step directions are always descent directions for this merit function.

FIGURE 25.1

The function $$c(x) = \sqrt{1 + {x}^{2}}$$ and its quadratic approximation at x = 2.

A summary of the algorithm is shown in Figure 25.2.

FIGURE 25.2

The path-following method for convex programming problems.

7 Parting Words

A story is never over, but every book must have an end. So, we stop here mindful of the fact that there are many interesting things left unsaid and topics unexplored. We hope we have motivated the reader to pursue the story further without our assistance—by reading other books and/or research papers and even perhaps making his or her own contributions. Cheers.

Exercises

25.1

Piecewise Linear Approximation. Given real numbers $$b_{1} <b_{2} <\ldots <b_{k}$$, let f be a continuous function on $$\mathbb{R}$$ that is linear on each interval $$[b_{i},b_{i+1}]$$, i = 0, 1, …, k (for convenience we let $$b_{0} = -\infty$$ and $$b_{k+1} = \infty$$). Such a function is called piecewise linear and the numbers b i are called breakpoints. Piecewise linear functions are often used to approximate (continuous) nonlinear functions. The purpose of this exercise is to show how and why.

(a)

Every piecewise linear function can be written as a sum of a constant plus a linear term plus a sum of absolute value terms:

Let c i denote the slope of f on the interval [b i , b i + 1]. Derive an explicit expression for each of the a j ’s (including a ₀) in terms of the c i ’s.

(b)

In terms of the c i ’s, give necessary and sufficient conditions for f to be convex.

(c)

In terms of the a j ’s, give necessary and sufficient conditions for f to be convex.

(d)

Assuming that f is convex and is a term in the objective function for a linearly constrained optimization problem, derive an equivalent linear programming formulation involving at most k extra variables and constraints.

(e)

Repeat the first four parts of this problem using $$\max (x - b_{i},0)$$ in place of | x − b i  | .

25.2

Let f be the function of 2 real variables defined by

Show that f is convex.

25.3

A function f of 2 real variables is called a monomial if it has the form

for some nonnegative integers m and n. Which monomials are convex?

25.4

Let ϕ be a convex function of a single real variable. Let f be a function defined on $${\mathbb{R}}^{n}$$ by the formula

where a is an n-vector and b is a scalar. Show that f is convex.

25.5

Which of the following functions are convex (assume that the domain of the function is all of $${\mathbb{R}}^{n}$$ unless specified otherwise)?

(a)

(b)

(c)

(d)

(e)

(f)

(g)

$$\frac{{x}^{2}} {y}$$ on $$\{(x,y) : y> 0\}$$

25.6

Given a symmetric square matrix A, the quadratic form $${x}^{T}Ax =\sum _{i,j}$$ $$a_{ij}x_{i}x_{j}$$ generalizes the notion of the square of a variable. The generalization of the notion of the fourth power of a variable is an expression of the form

The four-dimensional array of numbers $$A =\{ a_{ijkl} : 1 \leq i \leq n,1 \leq j \leq n,1 \leq k \leq n,1 \leq l \leq n\}$$ is called a 4-tensor. As with quadratic expressions, we may assume that A is symmetric:

(i.e., given i, j, k, l, all 4! = 24 premutations must give the same value for the tensor).

(a)

Give conditions on the 4-tensor A to guarantee that f is convex.

(b)

Suppose that some variables, say y i ’s, are related to some other variables, say x j ’s, in a linear fashion:

Express $$\sum _{i}y_{i}^{4}$$ in terms of the x j ’s. In particular, give an explicit expression for the 4-tensor and show that it satisfies the conditions derived in part (a).

25.7

Consider the problem

where − 1 < a < 1.

(a)

Graph the feasible set: $$\left \{(x_{1},x_{2}) : \sqrt{{\epsilon }^{2 } + x_{1 }^{2}} \leq x_{2}\right \}$$. Is the problem convex?

(b)

Following the steps in the middle of p. 391 of the text, write down the first-order optimality conditions for the barrier problem associated with barrier parameter μ > 0.

(c)

Solve explicitly the first-order optimality conditions. Let $$(x_{1}(\mu ),x_{2}(\mu ))$$ denote the solution.

(d)

Graph the central path, (x ₁(μ), x ₂(μ)), as μ varies from 0 to ∞.

25.8

Multidimensional Taylor’s series expansion. Given a function g(t) defined for real values of t, the three-term Taylor’s series expansion with remainder is

The remainder term satisfies

Let f be a smooth function defined on $${\mathbb{R}}^{n}$$. Apply the three-term Taylor’s series expansion to $$g(t) = f(x + t\Delta x)$$ to show that

25.9

Consider the following convex programming problem:

(a)

Find the quadratic subproblem if the current primal solution is $$(\bar{x}_{1},\bar{x}_{2}) = (1/2,-2/3)$$ and the current dual solution is $$\bar{y} = 2$$.

(b)

Show that for arbitrary current primal and dual solutions, the feasible set for the convex programming problem is contained within the feasible set for the quadratic approximation.

Notes

Interior-point methods for nonlinear programming can be traced back to the pioneering work of Fiacco and McCormick (1968). For more on interior-point methods for convex programming, see Nesterov and Nemirovsky (1993) or den Hertog (1994).

The fact that the step directions are descent directions for the merit function Ψ is proved in Vanderbei and Shanno (1999).

Bibliography

den Hertog, D. (1994). Interior point approach to linear, quadratic, and convex programming. Dordrecht: Kluwer.Crossref

Fiacco, A., and McCormick, G. (1968). Nonlinear programming: Sequential unconstrained minimization techniques. McLean: Research Analysis Corporation. Republished in 1990 by SIAM, Philadelphia.

Nesterov, Y., and Nemirovsky, A. (1993). Interior point polynomial methods in convex programming: Theory and algorithms. Philadelphia: SIAM.

Vanderbei, R., and Shanno, D. (1999). http://​www.​sor.​princeton.​edu/​~rvdb/​ps/​nonlin.​pdfAn interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13, 231–252.

© Springer Science+Business Media New York 2017

Robert J. VanderbeiLinear ProgrammingInternational Series in Operations Research & Management Science196https://doi.org/10.1007/978-1-4614-7630-6_26

Erratum

Erratum

Robert J. Vanderbei1 

(1)

ORFE Department, Princeton University, Princeton, New Jersey, USA

The updated original online version for this book can be found at DOI 10.​1007/​978-1-4614-7630-6

The publisher regrets the errors published in the print and online versions of this book, corrections to Chapter 3, page 29, and Chapter 14, page 211, have been updated and can be found on the next pages.

Chapter 3

Degeneracy

Page 29, the below three display equations were wrong

should be replaced by

The updated original online version for this chapter can be found at DOI 10.​1007/​978-1-4614-7630-6_​3

Chapter 14

Network Flow Problems

Page 211, “Hence, arc (f,c) must enter the spanning tree” should read “Hence, arc (f,b) must enter the spanning tree”.

The updated original online version for this chapter can be found at DOI 10.​1007/​978-1-4614-7630-6_​14

Appendix A Source Listings

The algorithms presented in this book have all been implemented and are publicly available from the author’s web site:

http://​www.​princeton.​edu/​~rvdb/​LPbook/​

There are two variants of the simplex method: the two-phase method as shown in Figure 6.​1 and the self-dual method as shown in Figure 7.​1. The simplex codes require software for efficiently solving basis systems. There are two options: the eta-matrix approach described in Section 8.3 and the refactorization approach described in Section 8.5. Each of these “engines” can be used with either simplex method. Hence, there are in total four possible simplex codes that one can experiment with.

There are three variants of interior-point methods: the path-following method as shown in Figure 18.​1, the homogeneous self-dual method shown in Figure 22.​1 (modified to take long steps), and the long-step homogeneous self-dual method described in Exercise 22.4 of Chapter 22.

The source code that implements the algorithms mentioned above share as much common code as possible. For example, they all share the same input and output routines (the input routine, by itself, is a substantial piece of code). They also share code for the common linear algebra functions. Therefore, the difference between two methods is limited primarily to the specific function that implements the method itself.

The total number of lines of code used to implement all of the algorithms is about 9,000. That is too many lines to reproduce all of the code here. But the routines that actually lay out the particular algorithms are fairly short, only about 300 lines each. The relevant part of the self-dual simplex method is shown starting on the next page. It is followed by a listing of the relevant part of the homogeneous self-dual method.

The Self-Dual Simplex Method

    /****************************************************************

    *   Main loop                                                   *

    ****************************************************************/

    for (iter=0; iter<MAX_ITER; iter++) {

      /*************************************************************

      * STEP 1: Find mu                                            *

      *************************************************************/

      mu = -HUGE_VAL;

      col_in  = -1;

      for (j=0; j<n; j++) {

                if (zbar_N[j] > EPS2) {

                        if ( mu < -z_N[j]/zbar_N[j] ) {

                             mu = -z_N[j]/zbar_N[j];

                             col_in  = j;

                        }

                }

      }

      col_out = -1;

      for (i=0; i<m; i++) {

                if (xbar_B[i] > EPS2) {

                        if ( mu < -x_B[i]/xbar_B[i] ) {

                             mu = -x_B[i]/xbar_B[i];

                             col_out = i;

                             col_in  = -1;

                        }

                }

      }

      if ( mu <= EPS3 ) {       /* OPTIMAL */

          status = 0;

          break;

      }

      if ( col_out >= 0 ) {

        /*************************************************************

        *                          -1  T                             *

        * STEP 2: Compute dz  = -(B  N) e                            *

        *                   N            i                           *

        *         where i = col_out                                  *

        *************************************************************/

        vec[0] = -1.0;

        ivec[0] = col_out;

        nvec = 1;

        btsolve( m, vec, ivec, &nvec );

        Nt_times_z( N, at, iat, kat, basicflag, vec, ivec, nvec,

                     dz_N, idz_N, &ndz_N );

        /*************************************************************

        * STEP 3: Ratio test to find entering column                 *

        *************************************************************/

        col_in = ratio_test( dz_N, idz_N, ndz_N, z_N, zbar_N, mu );

        if (col_in == -1) {     /* INFEASIBLE */

            status = 2;

            break;

        }

        /*************************************************************

        *                        -1                                  *

        * STEP 4: Compute dx  = B  N e                               *

        *                   B         j                              *

        *                                                            *

        *************************************************************/

        j = nonbasics[col_in];

        for (i=0, k=ka[j]; k<ka[j+1]; i++, k++) {

             dx_B[i] =  a[k];

            idx_B[i] = ia[k];

        }

        ndx_B = i;

        bsolve( m, dx_B, idx_B, &ndx_B );

      } else {

        /*************************************************************

        *                        -1                                  *

        * STEP 2: Compute dx  = B  N e                               *

        *                   B         j                              *

        *************************************************************/

        j = nonbasics[col_in];

        for (i=0, k=ka[j]; k<ka[j+1]; i++, k++) {

             dx_B[i] =  a[k];

            idx_B[i] = ia[k];

        }

        ndx_B = i;

        bsolve( m, dx_B, idx_B, &ndx_B );

        /*************************************************************

        * STEP 3: Ratio test to find leaving column                  *

        *************************************************************/

        col_out = ratio_test( dx_B, idx_B, ndx_B, x_B, xbar_B, mu );

        if (col_out == -1) {    /* UNBOUNDED */

            status = 1;

            break;

        }

        /*************************************************************

        *                          -1  T                             *

        * STEP 4: Compute dz  = -(B  N) e                            *

        *                   N            i                           *

        *                                                            *

        *************************************************************/

         vec[0] = -1.0;

        ivec[0] = col_out;

        nvec = 1;

        btsolve( m, vec, ivec, &nvec );

        Nt_times_z( N, at, iat, kat, basicflag, vec, ivec, nvec,

                     dz_N, idz_N, &ndz_N );

      }

      /*************************************************************

      *                                                            *

      * STEP 5: Put       t = x /dx                                *

      *                        i   i                               *

      *                   _   _                                    *

      *                   t = x /dx                                *

      *                        i   i                               *

      *                   s = z /dz                                *

      *                        j   j                               *

      *                   _   _                                    *

      *                   s = z /dz                                *

      *                        j   j                               *

      *************************************************************/

      for (k=0; k<ndx_B; k++) if (idx_B[k] == col_out) break;

      t    =    x_B[col_out]/dx_B[k];

      tbar = xbar_B[col_out]/dx_B[k];

      for (k=0; k<ndz_N; k++) if (idz_N[k] == col_in) break;

      s    =    z_N[col_in]/dz_N[k];

      sbar = zbar_N[col_in]/dz_N[k];

      /*************************************************************

      *                                _    _    _                 *

      * STEP 7: Set z  = z  - s dz     z  = z  - s dz              *

      *              N    N       N     N    N       N             *

      *                                _    _                      *

      *             z  = s             z  = s                      *

      *              i                  i                          *

      *             _    _    _                                    *

      *             x  = x  - t dx     x  = x  - t dx              *

      *              B    B       B     B    B       B             *

      *             _    _                                         *

      *             x  = t             x  = t                      *

      *              j                  j                          *

      *************************************************************/

      for (k=0; k<ndz_N; k++) {

                j = idz_N[k];

                z_N[j]    -= s   *dz_N[k];

                zbar_N[j] -= sbar*dz_N[k];

      }

      z_N[col_in]    = s;

      zbar_N[col_in] = sbar;

      for (k=0; k<ndx_B; k++) {

                i = idx_B[k];

                x_B[i]    -= t   *dx_B[k];

                xbar_B[i] -= tbar*dx_B[k];

      }

      x_B[col_out]     = t;

      xbar_B[col_out]  = tbar;

      /*************************************************************

      * STEP 8: Update basis                                       *

      *************************************************************/

      i =    basics[col_out];

      j = nonbasics[col_in];

      basics[col_out]   = j;

      nonbasics[col_in] = i;

      basicflag[i] = -col_in-1;

      basicflag[j] = col_out;

      /*************************************************************

      * STEP 9: Refactor basis and print statistics                *

      *************************************************************/

      from_scratch = refactor( m, ka, ia, a, basics, col_out, v );

      if (from_scratch) {

          primal_obj = sdotprod(c,x_B,basics,m) + f;

          printf("%8d   %14.7e %9.2e \n", iter, primal_obj, mu );

          fflush(stdout);

      }

    }

The Homogeneous Self-Dual Method

    /****************************************************************

    *   Main loop                                                   *

    ****************************************************************/

    for (iter=0; iter<MAX_ITER; iter++) {

        /*************************************************************

        * STEP 1: Compute mu and centering parameter delta.

        *************************************************************/

        mu = (dotprod(z,x,n)+dotprod(w,y,m)+phi*psi) / (n+m+1);

        if (iter%2 == 0) {

            delta = 0.0;

        } else {

            delta = 1.0;

        }

        /*************************************************************

        * STEP 1: Compute primal and dual objective function values.

        *************************************************************/

        primal_obj = dotprod(c,x,n);

        dual_obj   = dotprod(b,y,m);

        /*************************************************************

        * STEP 2: Check stopping rule.

        *************************************************************/

        if ( mu < EPS ) {

            if ( phi > EPS ) {

                status = 0;

                break;       /* OPTIMAL */

            }

            else

            if ( dual_obj < 0.0) {

                status = 2;

                break;       /* PRIMAL INFEASIBLE */

            }

            else

            if ( primal_obj > 0.0) {

                status = 4;

                break;       /* DUAL INFEASIBLE */

            }

            else

            {

                status = 7;  /* NUMERICAL TROUBLE */

                break;

            }

        }

        /*************************************************************

        * STEP 3: Compute infeasibilities.

        *************************************************************/

        smx(m,n,A,kA,iA,x,rho);

        for (i=0; i<m; i++) {

            rho[i] = rho[i] - b[i]*phi + w[i];

        }

        normr = sqrt( dotprod(rho,rho,m) )/phi;

        for (i=0; i<m; i++) {

            rho[i] = -(1-delta)*rho[i] + w[i] - delta*mu/y[i];

        }

        smx(n,m,At,kAt,iAt,y,sigma);

        for (j=0; j<n; j++) {

            sigma[j] = -sigma[j] + c[j]*phi + z[j];

        }

        norms = sqrt( dotprod(sigma,sigma,n) )/phi;

        for (j=0; j<n; j++) {

            sigma[j] = -(1-delta)*sigma[j] + z[j] - delta*mu/x[j];

        }

        gamma = -(1-delta)*(dual_obj - primal_obj + psi) + psi - delta*mu/phi;

        /*************************************************************

        * Print statistics.

        *************************************************************/

        printf("%8d   %14.7e  %8.1e    %14.7e  %8.1e  %8.1e \n",

                iter, primal_obj/phi+f, normr,

                      dual_obj/phi+f,   norms, mu );

        fflush(stdout);

        /*************************************************************

        * STEP 4: Compute step directions.

        *************************************************************/

        for (j=0; j<n; j++) { D[j] = z[j]/x[j]; }

        for (i=0; i<m; i++) { E[i] = w[i]/y[i]; }

        ldltfac(n, m, kAt, iAt, At, E, D, kA, iA, A, v);

        for (j=0; j<n; j++) { fx[j] = -sigma[j]; }

        for (i=0; i<m; i++) { fy[i] =  rho[i]; }

        forwardbackward(E, D, fy, fx);

        for (j=0; j<n; j++) { gx[j] = -c[j]; }

        for (i=0; i<m; i++) { gy[i] = -b[i]; }

        forwardbackward(E, D, gy, gx);

        dphi = (dotprod(c,fx,n)-dotprod(b,fy,m)+gamma)/

               (dotprod(c,gx,n)-dotprod(b,gy,m)-psi/phi);

        for (j=0; j<n; j++) { dx[j] = fx[j] - gx[j]*dphi; }

        for (i=0; i<m; i++) { dy[i] = fy[i] - gy[i]*dphi; }

        for (j=0; j<n; j++) { dz[j] = delta*mu/x[j] - z[j] - D[j]*dx[j]; }

        for (i=0; i<m; i++) { dw[i] = delta*mu/y[i] - w[i] - E[i]*dy[i]; }

        dpsi = delta*mu/phi - psi - (psi/phi)*dphi;

        /*************************************************************

        * STEP 5: Compute step length (long steps).

        *************************************************************/

        theta = 0.0;

        for (j=0; j<n; j++) {

            if (theta < -dx[j]/x[j]) { theta = -dx[j]/x[j]; }

            if (theta < -dz[j]/z[j]) { theta = -dz[j]/z[j]; }

        }

        for (i=0; i<m; i++) {

            if (theta < -dy[i]/y[i]) { theta = -dy[i]/y[i]; }

            if (theta < -dw[i]/w[i]) { theta = -dw[i]/w[i]; }

        }

        theta = MIN( 0.95/theta, 1.0 );

        /*************************************************************

        * STEP 6: Step to new point

        *************************************************************/

        for (j=0; j<n; j++) {

            x[j] = x[j] + theta*dx[j];

            z[j] = z[j] + theta*dz[j];

        }

        for (i=0; i<m; i++) {

            y[i] = y[i] + theta*dy[i];

            w[i] = w[i] + theta*dw[i];

        }

        phi = phi + theta*dphi;

        psi = psi + theta*dpsi;

    }

Answers to Selected Exercises

1.3

See Exercise 2.19.

2.1

$$(x_{1 } , x_{2 } , x_{3},x_{4}) = (2,0,1,0)$$, ζ = 17.

2.2

$$(x_{1 } , x_{2 } ) = (1,0)$$, ζ = 2.

2.3

$$(x_{1 } , x_{2 } , x_{3}) = (0,0.5,1.5)$$, $$\zeta = -3$$.

2.4

$$(x_{1 } , x_{2 } , x_{3}) = (0,1,0)$$, $$\zeta = -3$$.

2.5

$$(x_{1 } , x_{2 } ) = (2,1)$$, ζ = 5.

2.6

Infeasible.

2.7

Unbounded.

2.8

$$(x_{1 } , x_{2 } ) = (4,8)$$, ζ = 28.

2.9

$$(x_{1 } , x_{2 } , x_{3}) = (1.5,2.5,0)$$, ζ = 10. 5.

2.10

$$(x_{1 } , x_{2 } , x_{3},x_{4}) = (0,0,0,1)$$, ζ = 9.

2.11

$$(x_{12 } , x_{13},x_{14},x_{23},x_{24},x_{34}) = (1,0,0,1,0,1)$$, ζ = 6.

7.1

(1) x  ∗  = (2, 4, 0, 0, 0, 0, 8), ξ  ∗  = 14. (2) x  ∗  unchanged, ξ  ∗  = 12. 2. (3) x  ∗  = (0, 8, 0, 0, 0, 10, 10), ξ  ∗  = 16.

7.2

Δc ₁ ∈ ( − ∞, 1. 2], Δc ₂ ∈ [ − 1. 2, ∞), Δc ₃ ∈ [ − 1, 9], Δc ₄ ∈ ( − ∞, 2. 8].

9.1

$$(x_{1 } , x_{2 } ) = (0,5)$$.

9.2

$$(x_{1 } , x_{2 } , x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}) = (0,6,1,15,2,1,0,0)$$.

10.5

The fundamental theorem was proved only for problems in standard form. The LP here can be reduced to standard form.

11.1

A should hide a or b with probabilities $$b/(a + b)$$ and $$a/(a + b)$$, respectively. B should hide a or b with equal probability.

11.3

12.1

Slope = 2 ∕ 7, intercept = 1.

12.2

Slope = 1 ∕ 2, intercept = 0.

12.7

(2) 340. (3) x  ∗  is chosen so that the number of months in which extra workers will be used is equal to the number of months in the cycle (12) times the inhouse employee cost ($17.50) divided by the outhouse employee cost ($25) rounded down to the nearest integer.

12.8

Using L ¹, g = 8. 951. With L ², g = 8. 924.

13.1

μ

Bonds

Materials

Energy

Financial

5.0000–

0.0000

1.0000

0.0000

0.0000

1.9919–5.0000

0.0000

0.9964

0.0036

0.0000

1.3826–1.9919

0.0000

0.9335

0.0207

0.0458

0.7744–1.3826

0.0000

0.9310

0.0213

0.0477

0.5962–0.7744

0.0000

0.7643

0.0666

0.1691

0.4993–0.5962

0.6371

0.2764

0.0023

0.0842

0.4659–0.4933

0.6411

0.2733

0.0019

0.0836

0.4548–0.4659

0.7065

0.2060

0.0000

0.0875

0.4395–0.4548

0.7148

0.1966

0.0000

0.0886

0.2606–0.4395

0.8136

0.0952

0.0000

0.0912

0.0810–0.2606

0.8148

0.0939

0.0000

0.0913

0.0000–0.0810

0.8489

0.0590

0.0000

0.0922

13.1

μ

Hair

Cosmetics

Cash

3.5–

1.0

0.0

0.0

1.0–3.5

0.7

0.3

0.0

0.5–1.0

0.5

0.5

0.0

0.0–0.5

0.0

0.0

1.0

14.6

The optimal spanning tree consists of the following arcs:

The solution is not unique.

17.1

17.2

Let c = cosθ. If c≠ 0, then $$x_{1 } = \left (c - 2\mu + \sqrt{{c}^{2 } + {4\mu }^{2}}\right )/2c$$, else $$x_{1 } = 1/2$$. Formula for x ₂ is the same except that cos θ is replaced by $$\sin \theta$$.

17.3

$$\max \{{c}^{T } x +\sum _{j } r_{j}\log x_{j} +\sum _{i}s_{i}\log w_{i} : Ax \leq b,x \geq 0\}$$.

18.1

Using $$\delta = 1/10$$ and $$r = 9/10$$:

1.

2.

3.

4.

20.1

20.3

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Index

AA

LDL T −factorization

Lp−norm

L²−regression

LU-factorization

A

Acyclic network

Adler and Berenguer (1981)

Adler and Megiddo (1985)

Adler et al. (1989)

Affine-scaling algorithm

Ahuja et al. (1993)

Algorithm

affine-scaling

Dijkstra’s

dual simplex

homogeneous, self-dual interior-point

parametric self-dual simplex

path-following

convex programming

general form

quadratic programming

primal simplex

self-dual network simplex

self-dual parametric simplex

successive quadratic programming

Anstreicher (1996)

Arbitrage

Arc

head of

Ascent direction

Assignment problem

Auxiliary problem

Average-case analysis

Axes, pitch, roll, and yaw

B

Backward substitution

Balanced flow

Barnes (1986)

Barrier function

Barrier problem

Basic feasible solution

Basic variables

Bayer and Lagarias (1989a)

Bayer and Lagarias (1989b)

Bazaraa et al. (1977)

Bellman (1957)

Bellman’s equation

Bendsøe et al. (1994)

Ben-Tal, A.

Bernstein, D.H.

Bertsekas (1991)

Bertsekas (1995)

Bimatrix games

Bipartite graph

Bland (1977)

Bland’s rule

Bloomfield and Steiger (1983)

Bluffing

Borgwardt (1982)

Borgwardt (1987a)

Borgwardt (1987b)

Bradley et al. (1977)

Branch-and-bound

Breadth-first search

Bump

C

Capacity

cut

Carathéodory (1907)

Carpenter et al. (1993)

Central path

Centroid

Certificate of optimality

Charnes (1952)

Christofides (1975)

Chvátal (1983)

Çınlar, E.

Column dominance

Come-from

Complementarity

strict

Complementary slackness theorem

Complexity

predictor-corrector algorithm

Complexity theory

Concave function

Connected

Connected components

Conservation laws

Continuous paths

Convergence

Convex analysis

Convex combination

Convex function

Convex hull

Convex programming

interior-point method for

Convex quadratic programming problem

Convex set

Cook (2012)

Corrector step

Cost

lost opportunity

Crew scheduling problem

Critical point(s)

Cut set

Cycle

Cycling

D

Dantzig (1951a)

Dantzig (1951b)

Dantzig (1955)

Dantzig (1963)

Dantzig and Orchard-Hayes (1954)

Dantzig et al. (1955)

Dantzig, G.B.

Dantzig, T.

Decision variables

Degeneracy

geometry

Degenerate dictionary

Degenerate pivot

Demand node

den Hertog (1994)

Depth-first search

Destination node

Devex

Dictionary

Diet problem

Digraph

Dijkstra (1959)

Dijkstra’s algorithm

Dikin (1967)

Dikin (1974)

Directed paths

Dodge (1987)

Dorn et al. (1964)

Dresher (1961)

Dual estimates

Duality theory

Dual network simplex method

Dual pivot

Dual problem

general form

Dual simplex method

Duff et al. (1986)

Dynamic programming

E

Edge

Efficiency

simplex method

Efficient frontier

Elias et al. (1956)

Ellipsoid method

Entering arc

Entering variable

Enumeration tree

Equilibrium

Nash

Equivalence classes

Equivalence relation

Eta-file

Eta matrix

F

Facet

Facility location

Factorization

LDL T

LU

instability and quasidefinite matrices

stability and positive definite matrices

Fair game

Farkas (1902)

Farkas’ lemma

Feasible flow

Feasible solution

Fiacco and McCormick (1968)

Fixed costs

Flow balance constraints

Folven, G.

Ford and Fulkerson (1956)

Ford and Fulkerson (1958)

Ford and Fulkerson (1962)

Forrest and Tomlin (1972)

Forward substitution

Fourer and Mehrotra (1991)

Fourer et al. (1993)

Fourer, R.

Fourier

Fulkerson and Dantzig (1955)

Function

barrier

concave

strongly

convex

strongly

objective

G

Gal (1993)

Gale et al. (1951)

Games

bimatrix

zero-sum

Game theory

Garey and Johnson (1977)

Garfinkel and Nemhauser (1972)

Garfinkel, R.S.

Gass and Saaty (1955)

Gay (1985)

Gay, D.M.

Gill et al. (1991)

Gill et al. (1992)

Gilmartin, J.

Goldfarb and Reid (1977)

Golub and VanLoan (1989)

Gonin and Money (1989)

Gordan (1873)

Go-to

Gradient

Graph

Gravity, acceleration due to

Gross, L.

H

Halfspace

generalized

Hall and Vanderbei (1993)

Hall, L.A.

Harris (1973)

Hedging

Hemp (1973)

Hessian

Higher-order methods

Hillier and Lieberman (1977)

Hitchcock (1941)

Hitchcock transportation problem

Hoffman (1953)

H¨older’s inequality

Homogeneous problem

Homogeneous self-dual method

Homotopy method

Howard (1960)

Huard (1967)

I

Ikura, Y.

Incidence matrix

Infeasibility

Infeasible dictionary

Initialization

dual-based

Integer programming

Integer programming problem

Integrality theorem

Interior-point methods

affine-scaling

homogeneous, self-dual

path-following

convex programming

general form

quadratic programming

Inventory, cost of

Iteratively reweighted least squares

Iterative reweighted least squares

J

Jensen and Barnes (1980)

John (1948)

Joint

K

Kantorovich (1960)

Karlin (1959)

Karmarkar (1984)

Karmarkar, N.K.

Karush (1939)

Karush–Kuhn–Tucker (KKT) system

Karush, W.

Kennington and Helgason (1980)

Kernighan, B.

Khachian (1979)

Klee and Minty (1972)

Klee, V.

Knapsack problem

Kojima et al. (1989)

König’s theorem

Koopmans, T.C.

Kotzig (1956)

Kuhn (1950)

Kuhn (1976)

Kuhn and Tucker (1951)

Kuhn, H.W.

L

Label-correcting algorithm

Labels

Label-setting algorithm

Lagrange multipliers

Lagrangian

Lagrangian duality

Lawler (1976)

Least squares

Leaving arc

Leaving variable

Legendre transform

Legs

Lemke (1954)

Lemke (1965)

Lemma

Farkas’

Lexicographic method

Linear complementarity problem

Linear program

standard form

Logarithmic barrier function

Long-step method

Lower bounds

LP-relaxation

Luenberger (1984)

Lustig (1990)

Lustig et al. (1994)

Lustig, I.J.

M

Mandelbaum, A.

Markowitz (1957)

Markowitz (1959)

Markowitz model

Marshall and Suurballe (1969)

Mascarenhas (1997)

Matrix

notation

positive definite

quasidefinite

sparse

Matrix game

Maximum-flow problem

Mean

Median

Mediocrity, measures thereof

Megiddo (1989)

Mehrotra (1989)

Mehrotra (1992)

Meketon, M.S.

Member

Method

affine-scaling

dual network simplex

dual simplex

higher-order

homogeneous self-dual

homogeneous, self-dual interior-point

homotopy

lexicographic

long-step

Newton’s

out-of-kilter

parametric self-dual simplex

path-following

convex programming

general form

quadratic programming

perturbation

predictor-corrector

primal–dual simplex

primal network simplex

primal simplex

self-dual network simplex

self-dual parametric simplex

short-step

simplex

two phase

Michell (1904)

Midrange

Minimum-cost network flow problem

Minimum-degree ordering

Minimum operator ^

Minimum-weight structural design problem

Mixed integer programming problem

Mizuno et al. (1993)

Moment arm

Monteiro and Adler (1989)

N

Nabona, N.

Nash and Sofer (1996)

Nash equilibrium

Nazareth (1986)

Nazareth (1987)

Nazareth (1996)

Negative transpose

Nemhauser and Wolsey (1988)

Nesterov and Nemirovsky (1993)

Network

acyclic

complete

connected

Network flow problems

Network simplex method

Newton’s method

Node

Node-arc incidence matrix

Nonbasic variables

Nonlinear objective function

Nonlinear programming

Normal equations

Null space

Null variable

O

Objective function

Onion skin

Optimality

check for

Optimality conditions

first-order

Optimal solution

Option pricing

Orden, A.

Orlin, J.B.

Orthogonal projection

Out-of-Kilter method

P

Parametric analysis

Parametric self-dual simplex method

Partial pricing

Path

Path-following method

Penalty function

Perturbation method

Phase I

dual-based

Phase II

Piecewise linear function

Pivot

Pivot rule

Planar network

Poker

Polyhedron

Polynomial algorithm

Polynomial complexity

Portfolio optimization

Portfolio selection

Positive definite matrix

Positive semidefinite matrix

closure under inversion

closure under summation

Predictor-corrector algorithm

complexity

Predictor step

Primal–dual network simplex method

Primal–dual scaling

Primal–dual simplex method

Primal flow

Primal network simplex method

Primal pivot

Primal problem

Primal scaling

Primal simplex method

Principle stresses

Probability

of infeasibility

of unboundedness

Problem

assignment

auxiliary

barrier

convex quadratic programming

crew scheduling

diet

dual

equipment scheduling

Hitchcock transportation

homogeneous linear programming

homogeneous self-dual

integer programming

knapsack

linear complementarity

linear programming

general form

maximum-flow

minimum-cost network

minimum-weight structural design

mixed integer programming

network flows

primal

quadratic penalty

quadratic programming

scheduling

self-dual linear programming

separable quadratic programming

set-covering

set-partitioning

shortest path

transportation

transshipment

traveling salesman

upper-bounded network flow

Programming

nonlinear

Projected gradient direction

scaled

Projection

Projective geometry

Pruning

Q

Quadratic form

Quadratic penalty problem

Quadratic programming problem

convex

dual

interior-point method for

Quasidefinite matrix

R

Ranges

Recski (1989)

Reduced KKT system

Regression

Regression model

Reid (1982)

Resource allocation

Revised simplex method

Reward

Risk

Rockafellar (1970)

Root

of a function

Root node

Roses

Route

Row dominance

Row operations

Rozvany (1989)

Ruszczynski, A

Ruszczyński and Vanderbei (2003)

S

Saddle points

Saigal (1995)

Sailing

Sales force planning

Saunders (1973)

Scale invariance

Scaling matrices

Scheduling problem

Second-order conditions

Self-dual linear program

Self-dual parametric simplex method

Self-dual problem

Self-dual simplex method

Sensitivity analysis

Separable quadratic programming problem

Separation theorem

Set-covering problem

Set-partitioning problem

Sherman–Morrison–Woodbury formula

Shortest-path problem

Short-step method

Simplex method

average-case analysis

dual

geometry

initialization

dual-based

parametric self-dual

primal

primal–dual

revised

self-dual network

unboundedness

worst-case analysis

Sink node

Skew symmetric matrix

Slack variable

Smale (1983)

Solution

basic feasible

feasible

optimal

tree

Source node

Spanning tree

Sparse matrix

Sparsity

Spike column

Stability

Stable structure

Steepest ascent

Steepest edge

Steiner tree

Step direction

affine-scaling

dual

primal

Step direction decomposition

Step length

Stiemke (1915)

Stochastic vector

Strategy

randomized

Strict complementarity

Strictly positive vector (>)

Strong duality theorem

Strongly concave function

Strongly convex function

Structural optimization

Subnetwork

Substitution

backward

forward

Successive approximations

Successive quadratic programming algorithm

Supply node

Supremum norm

Symmetric games

T

Tableau

simplex

Tail of an arc

Taylor’s series

Tensor

Theorem

ascent direction

Bland’s rule

Carathéodory

central path

complementary slackness

convergence of affine-scaling

convergence of simplex method

convex hull

efficiency of interior-point method

integrality

König

lexicographic rule

linear programming, fundamental

local maximum

max-flow min-cut

minimax

optimality for convex quadratic programs

separating hyperplane

spanning tree

strict complementary slackness

strong duality

weak duality

Todd (1986)

Todd (1995)

Todd, M.J.

Topology

Transportation problem

Transshipment problem

Traveling salesman problem

Tree

solution

spanning

Triangle inequality

Truss

Tsuchiya and Muramatsu (1992)

Tucker (1956)

Tucker, A.W.

Turner (1991)

Two-phase methods

U

Unboundedness

Underbidding

Unitary matrix

Upper-bounded network flow problem

Upper bounds

V

Value

Value function

Vanderbei (1989)

Vanderbei (1994)

Vanderbei (1995)

Vanderbei (1999)

Vanderbei and Carpenter (1993)

Vanderbei and Shanno (1999)

Vanderbei et al. (1986)

Variable

basic

decision

entering

leaving

nonbasic

null

slack

Vector norms

Vehicle routing

Vertex

Ville (1938)

von Neumann (1928)

von Neumann and Morgenstern (1947)

von Neumann, J.

W

Warnie, J.

Weak duality theorem

Wolkowicz, H.

Woolbert, S.

Worst-case analysis

Wright (1996)

Wu, L.

X

Xu et al. (1993)

Y

Yang, B.

Ye et al. (1994)