Linear programming: Foundations and extensions

469 Pages·2008·4.162 MB·English

by  Robert Vanderbei

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
Antibiotics Used in Food Animals
Ching & Ng/ MARKOV CHAINS: Models, Algorithms and Applications
Li & Sun/ NONLINEAR INTEGER PROGRAMMING
Kaliszewski/ SOFT COMPUTING FOR COMPLEX MULTIPLE CRITERIA DECISION MAKING
Bouyssou et al/ EVALUATION AND DECISION MODELS WITH MULTIPLE CRITERIA: Stepping
stones for the analyst
Blecker & Friedrich/ MASS CUSTOMIZATION: Challenges and Solutions
Appa, Pitsoulis & Williams/ HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
Herrmann/ HANDBOOK OF PRODUCTION SCHEDULING
Axsäter/ INVENTORY CONTROL, 2nd Ed.
Hall/ PATIENT FLOW: Reducing Delay in Healthcare Delivery
Józefowska & Węglarz/ PERSPECTIVES IN MODERN PROJECT SCHEDULING
Tian & Zhang/ VACATION QUEUEING MODELS: Theory and Applications
Yan, Yin & Zhang/ STOCHASTIC PROCESSES, OPTIMIZATION, AND CONTROL THEORY
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
Yu/ TECHNOLOGY PORTFOLIO PLANNING AND MANAGEMENT: Practical Concepts and Tools
Kandiller/ PRINCIPLES OF MATHEMATICS IN OPERATIONS RESEARCH
Lee & Lee/ BUILDING SUPPLY CHAIN EXCELLENCE IN EMERGING ECONOMIES
Weintraub/ MANAGEMENT OF NATURAL RESOURCES: A Handbook of Operations Research Models,
Algorithms, and Implementations
Hooker/ INTEGRATED METHODS FOR OPTIMIZATION
Dawande et al/ THROUGHPUT OPTIMIZATION IN ROBOTIC CELLS
Friesz/ NETWORK SCIENCE, NONLINEAR SCIENCE and INFRASTRUCTURE SYSTEMS
Cai, Sha & Wong/ TIME-VARYING NETWORK OPTIMIZATION
Mamon & Elliott/ HIDDEN MARKOV MODELS IN FINANCE
del Castillo/ PROCESS OPTIMIZATION: A Statistical Approach
Józefowska/JUST-IN-TIME SCHEDULING: Models & Algorithms for Computer & Manufacturing
Systems
Yu, Wang & Lai/ FOREIGN-EXCHANGE-RATE FORECASTING WITH ARTIFICIAL NEURAL
NETWORKS
Beyer et al/ MARKOVIAN DEMAND INVENTORY MODELS
Shi & Olafsson/ NESTED PARTITIONS OPTIMIZATION: Methodology And Applications
Samaniego/ SYSTEM SIGNATURES AND THEIR APPLICATIONS IN ENGINEERING RELIABILITY
Kleijnen/ DESIGN AND ANALYSIS OF SIMULATION EXPERIMENTS
Førsund/ HYDROPOWER ECONOMICS
Kogan & Tapiero/ SUPPLY CHAIN GAMES: Operations Management and Risk Valuation

* 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

© 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.

Printed on acid-free paper.

9 8 7 6 5 4 3 2 1

springer.com

ThetextforthisbookwasformattedinTimes-RomanusingAMS-LATEX(whichisamacropackageforLeslieLamport’s
LATEX,whichitselfisamacropackageforDonaldKnuth’sTEXtextformattingsystem)andconvertedtopdfformatusing
PDFLATEX. ThefigureswereproducedusingMicroSoft’sPOWERPOINT andwereincorporatedintothetextaspdffiles
w iththemacropackageGRAPHICX.TEX.
To Krisadee,
Marisa and Diana
Contents
Preface xiii
Prefaceto2ndEdition xvii
Prefaceto3rdEdition xix
Part1. BasicTheory—TheSimplexMethodandDuality 1
Chapter1. Introduction 3
1. ManagingaProductionFacility 3
2. TheLinearProgrammingProblem 6
Exercises 8
Notes 10
Chapter2. TheSimplexMethod 13
1. AnExample 13
2. TheSimplexMethod 16
3. Initialization 19
4. Unboundedness 22
5. Geometry 22
Exercises 24
Notes 27
Chapter3. Degeneracy 29
1. DefinitionofDegeneracy 29
2. TwoExamplesofDegenerateProblems 29
3. ThePerturbation/LexicographicMethod 32
4. Bland’sRule 36
5. FundamentalTheoremofLinearProgramming 38
6. Geometry 39
Exercises 42
Notes 43
Chapter4. EfficiencyoftheSimplexMethod 45
vii
viii CONTENTS
1. PerformanceMeasures 45
2. MeasuringtheSizeofaProblem 45
3. MeasuringtheEfforttoSolveaProblem 46
4. Worst-CaseAnalysisoftheSimplexMethod 47
Exercises 52
Notes 53
Chapter5. DualityTheory 55
1. Motivation—FindingUpperBounds 55
2. TheDualProblem 57
3. TheWeakDualityTheorem 58
4. TheStrongDualityTheorem 60
5. ComplementarySlackness 66
6. TheDualSimplexMethod 68
7. ADual-BasedPhaseIAlgorithm 71
8. TheDualofaProbleminGeneralForm 73
9. ResourceAllocationProblems 74
10. LagrangianDuality 78
Exercises 79
Notes 87
Chapter6. TheSimplexMethodinMatrixNotation 89
1. MatrixNotation 89
2. ThePrimalSimplexMethod 91
3. AnExample 96
4. TheDualSimplexMethod 101
5. Two-PhaseMethods 104
6. NegativeTransposeProperty 105
Exercises 108
Notes 109
Chapter7. SensitivityandParametricAnalyses 111
1. SensitivityAnalysis 111
2. ParametricAnalysisandtheHomotopyMethod 115
3. TheParametricSelf-DualSimplexMethod 119
Exercises 120
Notes 124
Chapter8. ImplementationIssues 125
1. SolvingSystemsofEquations: LU-Factorization 126
2. ExploitingSparsity 130
3. ReusingaFactorization 136
4. PerformanceTradeoffs 140
CONTENTS ix
5. UpdatingaFactorization 141
6. ShrinkingtheBump 145
7. PartialPricing 146
8. SteepestEdge 147
Exercises 149
Notes 150
Chapter9. ProblemsinGeneralForm 151
1. ThePrimalSimplexMethod 151
2. TheDualSimplexMethod 153
Exercises 159
Notes 160
Chapter10. ConvexAnalysis 161
1. ConvexSets 161
2. Carathe´odory’sTheorem 163
3. TheSeparationTheorem 165
4. Farkas’Lemma 167
5. StrictComplementarity 168
Exercises 170
Notes 171
Chapter11. GameTheory 173
1. MatrixGames 173
2. OptimalStrategies 175
3. TheMinimaxTheorem 177
4. Poker 181
Exercises 184
Notes 187
Chapter12. Regression 189
1. MeasuresofMediocrity 189
2. MultidimensionalMeasures: RegressionAnalysis 191
3. L2-Regression 193
4. L1-Regression 195
5. IterativelyReweightedLeastSquares 196
6. AnExample: HowFastistheSimplexMethod? 198
7. WhichVariantoftheSimplexMethodisBest? 202
Exercises 203
Notes 208
Chapter13. FinancialApplications 211
1. PortfolioSelection 211
x CONTENTS
2. OptionPricing 216
Exercises 221
Notes 222
Part2. Network-TypeProblems 223
Chapter14. NetworkFlowProblems 225
1. Networks 225
2. SpanningTreesandBases 228
3. ThePrimalNetworkSimplexMethod 233
4. TheDualNetworkSimplexMethod 237
5. PuttingItAllTogether 240
6. TheIntegralityTheorem 243
Exercises 244
Notes 252
Chapter15. Applications 253
1. TheTransportationProblem 253
2. TheAssignmentProblem 255
3. TheShortest-PathProblem 256
4. Upper-BoundedNetworkFlowProblems 259
5. TheMaximum-FlowProblem 262
Exercises 264
Notes 269
Chapter16. StructuralOptimization 271
1. AnExample 271
2. IncidenceMatrices 273
3. Stability 274
4. ConservationLaws 276
5. Minimum-WeightStructuralDesign 279
6. AnchorsAway 281
Exercises 284
Notes 284
Part3. Interior-PointMethods 287
Chapter17. TheCentralPath 289
Warning: NonstandardNotationAhead 289
1. TheBarrierProblem 289
2. LagrangeMultipliers 292
3. LagrangeMultipliersAppliedtotheBarrierProblem 295
4. Second-OrderInformation 297