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  • LINEAR PROGRAMMING

    Foundations and Extensions Third Edition

  • Recent titles in the INTERNATIONAL SERIES IN OPERATIONS RESEARCH & MANAGEMENT SCIENCE Frederick S. Hillier, Series Editor, Stanford University Sethi, Yan & Zhang/ INVENTORY AND SUPPLY CHAIN MANAGEMENT WITH FORECAST

    UPDATES Cox/ QUANTITATIVE HEALTH RISK ANALYSIS METHODS: Modeling the Human Health Impacts of

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    APPLICATIONS IN FINANCIAL ENGINEERING, QUEUEING NETWORKS, AND MANUFACTURING SYSTEMS

    Saaty & Vargas/ DECISION MAKING WITH THE ANALYTIC NETWORK PROCESS: Economic, Political, Social & Technological Applications w. Benefits, Opportunities, Costs & Risks

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    * A list of the early publications in the series is at the end of the book *

  • LINEAR PROGRAMMING

    Foundations and Extensions Third Edition

    Robert J. Vanderbei

    Dept. of Operations Research and Financial Engineering Princeton University, USA

  • Robert J. Vanderbei Princeton University New Jersey, USA Series Editor: Fred Hillier Stanford University Stanford, CA, USA

    ISBN-13: 978-0-387-74387-5 (HB) ISBN-13: 978-0-387-74388-2 (e-book) Library of Congress Control Number: 2007932884

    Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com

    The text for this book was formatted in Times-Roman using AMS-L

    ATEX(which is a macro package for Leslie Lamport’s LATEX, which itself is a macro package for Donald Knuth’s TEXtext formatting system) and converted to pdf format using PDFLATEX. The figures were produced using MicroSoft’s POWERPOINT and were incorporated into the text as pdf files with the macro package GRAPHICX.TEX.

    © 2008 by Robert J. Vanderbei All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

  • To Krisadee,

    Marisa and Diana

  • Contents

    Preface xiii

    Preface to 2nd Edition xvii

    Preface to 3rd Edition xix

    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 6 Exercises 8 Notes 10

    Chapter 2. The Simplex Method 13 1. An Example 13 2. The Simplex Method 16 3. Initialization 19 4. Unboundedness 22 5. Geometry 22 Exercises 24 Notes 27

    Chapter 3. Degeneracy 29 1. Definition of Degeneracy 29 2. Two Examples of Degenerate Problems 29 3. The Perturbation/Lexicographic Method 32 4. Bland’s Rule 36 5. Fundamental Theorem of Linear Programming 38 6. Geometry 39 Exercises 42 Notes 43

    Chapter 4. Efficiency of the Simplex Method 45

    vii

  • viii CONTENTS

    1. Performance Measures 45 2. Measuring the Size of a Problem 45 3. Measuring the Effort to Solve a Problem 46 4. Worst-Case Analysis of the Simplex Method 47 Exercises 52 Notes 53

    Chapter 5. Duality Theory 55 1. Motivation—Finding Upper Bounds 55 2. The Dual Problem 57 3. The Weak Duality Theorem 58 4. The Strong Duality Theorem 60 5. Complementary Slackness 66 6. The Dual Simplex Method 68 7. A Dual-Based Phase I Algorithm 71 8. The Dual of a Problem in General Form 73 9. Resource Allocation Problems 74 10. Lagrangian Duality 78 Exercises 79 Notes 87

    Chapter 6. The Simplex Method in Matrix Notation 89 1. Matrix Notation 89 2. The Primal Simplex Method 91 3. An Example 96 4. The Dual Simplex Method 101 5. Two-Phase Methods 104 6. Negative Transpose Property 105 Exercises 108 Notes 109

    Chapter 7. Sensitivity and Parametric Analyses 111 1. Sensitivity Analysis 111 2. Parametric Analysis and the Homotopy Method 115 3. The Parametric Self-Dual Simplex Method 119 Exercises 120 Notes 124

    Chapter 8. Implementation Issues 125 1. Solving Systems of Equations: LU -Factorization 126 2. Exploiting Sparsity 130 3. Reusing a Factorization 136 4. Performance Tradeoffs 140

  • CONTENTS ix

    5. Updating a Factorization 141 6. Shrinking the Bump 145 7. Partial Pricing 146 8. Steepest Edge 147 Exercises 149 Notes 150

    Chapter 9. Problems in General Form 151 1. The Primal Simplex Method 151 2. The Dual Simplex Method 153 Exercises 159 Notes 160

    Chapter 10. Convex Analysis 161 1. Convex Sets 161 2. Carathéodory’s Theorem 163 3. The Separation Theorem 165 4. Farkas’ Lemma 167 5. Strict Complementarity 168 Exercises 170 Notes 171

    Chapter 11. Game Theory 173 1. Matrix Games 173 2. Optimal Strategies 175 3. The Minimax Theorem 177 4. Poker 181 Exercises 184 Notes 187

    Chapter 12. Regression 189 1. Measures of Mediocrity 189 2. Multidimensional Measures: Regression Analysis 191 3. L2-Regression 193 4. L1-Regression 195 5. Iteratively Reweighted Least Squares 196 6. An Example: How Fast is the Simplex Method? 198 7. Which Variant of the Simplex Method is Best? 202 Exercises 203 Notes 208

    Chapter 13. Financial Applications 211 1. Portfolio Selection 211

  • x CONTENTS

    2. Option Pricing 216 Exercises 221 Notes 222

    Part 2. Network-Type Problems 223

    Chapter 14. Network Flow Problems 225 1. Networks 225 2. Spanning Trees and Bases 228 3. The Primal Network Simplex Method 233 4. The Dual Network Simplex Method 237 5. Putting It All Together 240 6. The Integrality Theorem 243 Exercises 244 Notes 252

    Chapter 15. Applications 253 1. The Transportation Problem 253 2. The Assignment Problem 255 3. The Shortest-Path Problem 256 4. Upper-Bounded Network Flow Problems 259 5. The Maximum-Flow Problem 262 Exercises 264 Notes 269

    Chapter 16. Structural Optimization 271 1. An Example 271 2. Incidence Matrices 273 3. Stability 274 4. Conservation Laws 276 5. Minimum-Weight Structural Design 279 6. Anchors Away 281 Exercises 284 Notes 284

    Part 3. Interior-Point Methods 287

    Chapter 17. The Central Path 289 Warning: Nonstandard Notation Ahead 289 1. The Barrier Problem 289 2. Lagrange Multipliers 292 3. Lagrange Multipliers Applied to the Barrier Problem 295 4. Second-Order Information 297

  • CONTENTS xi

    5. Existence 297 Exercises 299 Notes 301

    Chapter 18. A Path-Following Method 303 1. Computing Step Directions 303 2. Newton’s Method 305 3. Estimating an Appropriate Value for the Barrier Parameter 306 4. Choosing the Step Length Parameter 307 5. Convergence Analysis 308 Exercises 314 Notes 318

    Chapter 19. The KKT System 319 1. The Reduced KKT System 319 2. The Normal Equations 320 3. Step Direction Decomposition 322 Exercises 325 Notes 325

    Chapter 20. Implementation Issues 327 1. Factoring Positive Definite Matrices 327 2. Quasidefinite Matrices 331 3. Problems in General Form 337 Exercises 342 Notes 342

    Chapter 21. The Affine-Scaling Method 345 1. The Steepest Ascent Direction 345 2. The Projected Gradient Direction 347 3. The Projected Gradient Direction with Scaling 349 4. Convergen