Linear Programming PDF
Preview Linear Programming
Linear Programming: Foundations and Extensions Robert J. Vanderbei DEPARTMENT OF OPERATIONS RESEARCH AND FINANCIAL ENGINEERING, PRINCETONUNIVERSITY,PRINCETON,NJ08544 E-mailaddress: [email protected] LINEARPROGRAMMING:FOUNDATIONSANDEXTENSIONS SecondEdition Copyright(cid:13)c2001byRobertJ.Vanderbei.Allrightsreserved.PrintedintheUnitedStatesofAmerica.Exceptaspermitted undertheUnitedStatesCopyrightActof1976,nopartofthispublicationmaybereproducedordistributedinanyformor byanymeans,orstoredinadatabaseorretrievalsystem,withoutthepriorwrittenpermissionofthepublisher. ISBN0-0000-0000-0 ThetextforthisbookwasformatedinTimes-RomanandthemathematicswasformatedinMichaelSpivak’sMathtimes usingAMS-LATEX(whichisamacropackageforLeslieLamport’sLATEX,whichitselfisamacropackageforDonald Knuth’sTEXtextformattingsystem)andconvertedfromdevice-independenttopostscriptformatusingDVIPS. Thefig- ureswereproducedusingSHOWCASEonaSiliconGraphics,Inc. workstationandwereincorporatedintothetextas encapsulatedpostscriptfileswiththemacropackagecalledPSFIG.TEX. To my parents, Howard and Marilyn, my dear wife, Krisadee, and the babes, Marisa and Diana Contents Preface xiii Prefaceto2ndEdition xvii 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 1. PerformanceMeasures 45 2. MeasuringtheSizeofaProblem 45 vii viii CONTENTS 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 5. UpdatingaFactorization 141 6. ShrinkingtheBump 145 CONTENTS ix 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 Part2. Network-TypeProblems 211 Chapter13. NetworkFlowProblems 213 1. Networks 213 x CONTENTS 2. SpanningTreesandBases 216 3. ThePrimalNetworkSimplexMethod 221 4. TheDualNetworkSimplexMethod 225 5. PuttingItAllTogether 228 6. TheIntegralityTheorem 231 Exercises 232 Notes 240 Chapter14. Applications 241 1. TheTransportationProblem 241 2. TheAssignmentProblem 243 3. TheShortest-PathProblem 244 4. Upper-BoundedNetworkFlowProblems 247 5. TheMaximum-FlowProblem 250 Exercises 252 Notes 257 Chapter15. StructuralOptimization 259 1. AnExample 259 2. IncidenceMatrices 261 3. Stability 262 4. ConservationLaws 264 5. Minimum-WeightStructuralDesign 267 6. AnchorsAway 269 Exercises 272 Notes 272 Part3. Interior-PointMethods 275 Chapter16. TheCentralPath 277 Warning: NonstandardNotationAhead 277 1. TheBarrierProblem 277 2. LagrangeMultipliers 280 3. LagrangeMultipliersAppliedtotheBarrierProblem 283 4. Second-OrderInformation 285 5. Existence 285 Exercises 287 Notes 289 Chapter17. APath-FollowingMethod 291 1. ComputingStepDirections 291 2. Newton’sMethod 293 3. EstimatinganAppropriateValuefortheBarrierParameter 294 CONTENTS xi 4. ChoosingtheStepLengthParameter 295 5. ConvergenceAnalysis 296 Exercises 302 Notes 306 Chapter18. TheKKTSystem 307 1. TheReducedKKTSystem 307 2. TheNormalEquations 308 3. StepDirectionDecomposition 310 Exercises 313 Notes 313 Chapter19. ImplementationIssues 315 1. FactoringPositiveDefiniteMatrices 315 2. QuasidefiniteMatrices 319 3. ProblemsinGeneralForm 325 Exercises 331 Notes 331 Chapter20. TheAffine-ScalingMethod 333 1. TheSteepestAscentDirection 333 2. TheProjectedGradientDirection 335 3. TheProjectedGradientDirectionwithScaling 337 4. Convergence 341 5. FeasibilityDirection 343 6. ProblemsinStandardForm 344 Exercises 345 Notes 346 Chapter21. TheHomogeneousSelf-DualMethod 349 1. FromStandardFormtoSelf-DualForm 349 2. HomogeneousSelf-DualProblems 350 3. BacktoStandardForm 360 4. SimplexMethodvsInterior-PointMethods 363 Exercises 367 Notes 368 Part4. Extensions 371 Chapter22. IntegerProgramming 373 1. SchedulingProblems 373 2. TheTravelingSalesmanProblem 375 3. FixedCosts 378 xii CONTENTS 4. NonlinearObjectiveFunctions 378 5. Branch-and-Bound 380 Exercises 392 Notes 393 Chapter23. QuadraticProgramming 395 1. TheMarkowitzModel 395 2. TheDual 399 3. ConvexityandComplexity 402 4. SolutionViaInterior-PointMethods 404 5. PracticalConsiderations 406 Exercises 409 Notes 411 Chapter24. ConvexProgramming 413 1. DifferentiableFunctionsandTaylorApproximations 413 2. ConvexandConcaveFunctions 414 3. ProblemFormulation 414 4. SolutionViaInterior-PointMethods 415 5. SuccessiveQuadraticApproximations 417 6. MeritFunctions 417 7. PartingWords 421 Exercises 421 Notes 423 AppendixA. SourceListings 425 1. TheSelf-DualSimplexMethod 426 2. TheHomogeneousSelf-DualMethod 429 AnswerstoSelectedExercises 433 Bibliography 435 Index 443
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