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L ECTURES ON S TOCHASTIC P ROGRAMMING M T ODELING AND HEORY Alexander Shapiro Georgia Institute of Technology Atlanta, Georgia Darinka Dentcheva Stevens Institute of Technology Hoboken, New Jersey Andrzej Ruszczyn´ski Rutgers University New Brunswick, New Jersey Society for Industrial Mathematical Programming Society and Applied Mathematics Philadelphia Philadelphia Copyright © 2009by the Society for Industrial and Applied Mathematics and the Mathematical Programming Society 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6thFloor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. Cover image appears courtesy of Julia Shapiro. Library of Congress Cataloging-in-Publication Data Shapiro, Alexander, 1949- Lectures on stochastic programming : modeling and theory / Alexander Shapiro, Darinka Dentcheva, Andrzej Ruszczyn´ski. p. cm. -- (MPS-SIAM series on optimization ; 9) Includes bibliographical references and index. ISBN 978-0-898716-87-0 1. Stochastic programming. I. Dentcheva, Darinka. II. Ruszczyn´ski, Andrzej P. III. Title. T57.79.S54 2009 519.7--dc22 2009022942 is a registered trademark. is a registered trademark. SPbook (cid:1) (cid:1) 2009/8/20 pagevii (cid:1) (cid:1) Contents ListofNotations xi Preface xiii 1 StochasticProgrammingModels 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Inventory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.1 TheNewsVendorProblem . . . . . . . . . . . . . . . . 1 1.2.2 ChanceConstraints . . . . . . . . . . . . . . . . . . . . 5 1.2.3 MultistageModels . . . . . . . . . . . . . . . . . . . . . 6 1.3 MultiproductAssembly . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Two-StageModel . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 ChanceConstrainedModel . . . . . . . . . . . . . . . . 10 1.3.3 MultistageModel . . . . . . . . . . . . . . . . . . . . . 12 1.4 PortfolioSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 StaticModel . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.2 MultistagePortfolioSelection . . . . . . . . . . . . . . . 16 1.4.3 DecisionRules . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 SupplyChainNetworkDesign . . . . . . . . . . . . . . . . . . . . . 22 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Two-StageProblems 27 2.1 LinearTwo-StageProblems . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.1 BasicProperties . . . . . . . . . . . . . . . . . . . . . . 27 2.1.2 TheExpectedRecourseCostforDiscreteDistributions . 30 2.1.3 TheExpectedRecourseCostforGeneralDistributions . . 32 2.1.4 OptimalityConditions . . . . . . . . . . . . . . . . . . . 38 2.2 PolyhedralTwo-StageProblems . . . . . . . . . . . . . . . . . . . . . 42 2.2.1 GeneralProperties . . . . . . . . . . . . . . . . . . . . . 42 2.2.2 ExpectedRecourseCost . . . . . . . . . . . . . . . . . . 44 2.2.3 OptimalityConditions . . . . . . . . . . . . . . . . . . . 47 2.3 GeneralTwo-StageProblems . . . . . . . . . . . . . . . . . . . . . . 48 2.3.1 ProblemFormulation,Interchangeability . . . . . . . . . 48 2.3.2 ConvexTwo-StageProblems . . . . . . . . . . . . . . . 49 2.4 Nonanticipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 vii (cid:1) (cid:1) (cid:1) (cid:1) SPbook (cid:1) (cid:1) 2009/8/20 pageviii (cid:1) (cid:1) viii Contents 2.4.1 ScenarioFormulation . . . . . . . . . . . . . . . . . . . 52 2.4.2 DualizationofNonanticipativityConstraints . . . . . . . 54 2.4.3 NonanticipativityDualityforGeneralDistributions . . . 56 2.4.4 ValueofPerfectInformation. . . . . . . . . . . . . . . . 59 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3 MultistageProblems 63 3.1 ProblemFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.1 TheGeneralSetting . . . . . . . . . . . . . . . . . . . . 63 3.1.2 TheLinearCase . . . . . . . . . . . . . . . . . . . . . . 65 3.1.3 ScenarioTrees . . . . . . . . . . . . . . . . . . . . . . . 69 3.1.4 AlgebraicFormulationofNonanticipativityConstraints . 71 3.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2.1 ConvexMultistageProblems . . . . . . . . . . . . . . . 76 3.2.2 OptimalityConditions . . . . . . . . . . . . . . . . . . . 77 3.2.3 DualizationofFeasibilityConstraints . . . . . . . . . . . 80 3.2.4 DualizationofNonanticipativityConstraints . . . . . . . 82 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4 OptimizationModelswithProbabilisticConstraints 87 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2 ConvexityinProbabilisticOptimization . . . . . . . . . . . . . . . . 94 4.2.1 GeneralizedConcavityofFunctionsandMeasures . . . . 94 4.2.2 ConvexityofProbabilisticallyConstrainedSets . . . . . 106 4.2.3 ConnectednessofProbabilisticallyConstrainedSets . . . 113 4.3 SeparableProbabilisticConstraints . . . . . . . . . . . . . . . . . . . 114 4.3.1 ContinuityandDifferentiabilityPropertiesof DistributionFunctions . . . . . . . . . . . . . . . . . . . 114 4.3.2 p-EfficientPoints . . . . . . . . . . . . . . . . . . . . . 115 4.3.3 OptimalityConditionsandDualityTheory . . . . . . . . 122 4.4 OptimizationProblemswithNonseparableProbabilisticConstraints. . 132 4.4.1 DifferentiabilityofProbabilityFunctionsandOptimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.4.2 ApproximationsofNonseparableProbabilistic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.5 Semi-infiniteProbabilisticProblems . . . . . . . . . . . . . . . . . . 144 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5 StatisticalInference 155 5.1 StatisticalPropertiesofSampleAverageApproximationEstimators . . 155 5.1.1 ConsistencyofSAAEstimators . . . . . . . . . . . . . . 157 5.1.2 AsymptoticsoftheSAAOptimalValue . . . . . . . . . . 163 5.1.3 SecondOrderAsymptotics . . . . . . . . . . . . . . . . 166 5.1.4 MinimaxStochasticPrograms . . . . . . . . . . . . . . . 170 5.2 StochasticGeneralizedEquations . . . . . . . . . . . . . . . . . . . . 174 5.2.1 ConsistencyofSolutionsoftheSAAGeneralized Equations . . . . . . . . . . . . . . . . . . . . . . . . . 175 (cid:1) (cid:1) (cid:1) (cid:1) SPbook (cid:1) (cid:1) 2009/8/20 pageix (cid:1) (cid:1) Contents ix 5.2.2 AsymptoticsofSAAGeneralizedEquationsEstimators . 177 5.3 MonteCarloSamplingMethods. . . . . . . . . . . . . . . . . . . . . 180 5.3.1 ExponentialRatesofConvergenceandSampleSize EstimatesintheCaseofaFiniteFeasibleSet . . . . . . . 181 5.3.2 SampleSizeEstimatesintheGeneralCase . . . . . . . . 185 5.3.3 FiniteExponentialConvergence . . . . . . . . . . . . . . 191 5.4 Quasi–MonteCarloMethods . . . . . . . . . . . . . . . . . . . . . . 193 5.5 Variance-ReductionTechniques . . . . . . . . . . . . . . . . . . . . . 198 5.5.1 LatinHypercubeSampling . . . . . . . . . . . . . . . . 198 5.5.2 LinearControlRandomVariablesMethod . . . . . . . . 200 5.5.3 ImportanceSamplingandLikelihoodRatioMethods . . . 200 5.6 ValidationAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.6.1 EstimationoftheOptimalityGap . . . . . . . . . . . . . 202 5.6.2 StatisticalTestingofOptimalityConditions . . . . . . . . 207 5.7 ChanceConstrainedProblems . . . . . . . . . . . . . . . . . . . . . . 210 5.7.1 MonteCarloSamplingApproach . . . . . . . . . . . . . 210 5.7.2 ValidationofanOptimalSolution . . . . . . . . . . . . . 216 5.8 SAAMethodAppliedtoMultistageStochasticProgramming . . . . . 220 5.8.1 StatisticalPropertiesofMultistageSAAEstimators . . . 221 5.8.2 ComplexityEstimatesofMultistagePrograms . . . . . . 226 5.9 StochasticApproximationMethod . . . . . . . . . . . . . . . . . . . 230 5.9.1 ClassicalApproach. . . . . . . . . . . . . . . . . . . . . 230 5.9.2 RobustSAApproach . . . . . . . . . . . . . . . . . . . . 233 5.9.3 MirrorDescentSAMethod . . . . . . . . . . . . . . . . 236 5.9.4 AccuracyCertificatesforMirrorDescentSASolutions . . 244 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 6 RiskAverseOptimization 253 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.2 Mean–RiskModels . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6.2.1 MainIdeasofMean–RiskAnalysis . . . . . . . . . . . . 254 6.2.2 Semideviations . . . . . . . . . . . . . . . . . . . . . . . 255 6.2.3 WeightedMeanDeviationsfromQuantiles . . . . . . . . 256 6.2.4 AverageValue-at-Risk . . . . . . . . . . . . . . . . . . . 257 6.3 CoherentRiskMeasures . . . . . . . . . . . . . . . . . . . . . . . . . 261 6.3.1 DifferentiabilityPropertiesofRiskMeasures . . . . . . . 265 6.3.2 ExamplesofRiskMeasures . . . . . . . . . . . . . . . . 269 6.3.3 LawInvariantRiskMeasuresandStochasticOrders . . . 279 6.3.4 RelationtoAmbiguousChanceConstraints . . . . . . . . 285 6.4 OptimizationofRiskMeasures . . . . . . . . . . . . . . . . . . . . . 288 6.4.1 DualizationofNonanticipativityConstraints . . . . . . . 291 6.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 295 6.5 StatisticalPropertiesofRiskMeasures . . . . . . . . . . . . . . . . . 300 6.5.1 AverageValue-at-Risk . . . . . . . . . . . . . . . . . . . 300 6.5.2 AbsoluteSemideviationRiskMeasure . . . . . . . . . . 301 6.5.3 VonMisesStatisticalFunctionals . . . . . . . . . . . . . 304 6.6 TheProblemofMoments . . . . . . . . . . . . . . . . . . . . . . . . 306 (cid:1) (cid:1) (cid:1) (cid:1) SPbook (cid:1) (cid:1) 2009/8/20 pagex (cid:1) (cid:1) x Contents 6.7 MultistageRiskAverseOptimization . . . . . . . . . . . . . . . . . . 308 6.7.1 ScenarioTreeFormulation. . . . . . . . . . . . . . . . . 308 6.7.2 ConditionalRiskMappings . . . . . . . . . . . . . . . . 315 6.7.3 RiskAverseMultistageStochasticProgramming . . . . . 318 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 7 BackgroundMaterial 333 7.1 OptimizationandConvexAnalysis . . . . . . . . . . . . . . . . . . . 334 7.1.1 DirectionalDifferentiability . . . . . . . . . . . . . . . . 334 7.1.2 ElementsofConvexAnalysis . . . . . . . . . . . . . . . 336 7.1.3 OptimizationandDuality . . . . . . . . . . . . . . . . . 339 7.1.4 OptimalityConditions . . . . . . . . . . . . . . . . . . . 346 7.1.5 PerturbationAnalysis . . . . . . . . . . . . . . . . . . . 351 7.1.6 Epiconvergence . . . . . . . . . . . . . . . . . . . . . . 357 7.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 7.2.1 ProbabilitySpacesandRandomVariables . . . . . . . . 359 7.2.2 ConditionalProbabilityandConditionalExpectation . . . 363 7.2.3 MeasurableMultifunctionsandRandomFunctions . . . . 365 7.2.4 ExpectationFunctions . . . . . . . . . . . . . . . . . . . 368 7.2.5 UniformLawsofLargeNumbers . . . . . . . . . . . . . 374 7.2.6 LawofLargeNumbersforRandomSetsand Subdifferentials . . . . . . . . . . . . . . . . . . . . . . 379 7.2.7 DeltaMethod . . . . . . . . . . . . . . . . . . . . . . . 382 7.2.8 ExponentialBoundsoftheLargeDeviationsTheory . . . 387 7.2.9 UniformExponentialBounds . . . . . . . . . . . . . . . 393 7.3 ElementsofFunctionalAnalysis . . . . . . . . . . . . . . . . . . . . 399 7.3.1 ConjugateDualityandDifferentiability . . . . . . . . . . 401 7.3.2 LatticeStructure . . . . . . . . . . . . . . . . . . . . . . 403 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 8 BibliographicalRemarks 407 Bibliography 415 Index 431 (cid:1) (cid:1) (cid:1) (cid:1) SPbook (cid:1) (cid:1) 2009/8/20 pagexi (cid:1) (cid:1) List of Notations :=,equalbydefinition,333 Rn,n-dimensionalspace,333 AT,transposeofmatrix(vector)A,333 A, domainoftheconjugateofriskmea- C(X),spaceofcontinuousfunctions,165 sureρ,262 C∗,polarofconeC,337 C ,thespaceofnonemptycompactsub- n C1(V,Rn),spaceofcontinuouslydiffer- setsofRn,379 entiablemappings,176 P, set of probability density functions, IFF,influencefunction,304 263 L⊥,orthogonalof(linear)spaceL,41 S ,setofcontactpoints,399 z O(1),genericconstant,188 b(k;α,N), cdf of binomial distribution, Op(·),term,382 214 Sε, the set of ε-optimal solutions of the d,distancegeneratingfunction,236 trueproblem,181 g+(x),right-hand-sidederivative,297 Vd(A),LebesguemeasureofsetA⊂Rd, cl(A),topologicalclosureofsetA,334 195 conv(C),convexhullofsetC,337 W1,∞(U),spaceofLipschitzcontinuous Corr(X,Y),correlationofXandY,200 functions,166,353 Cov(X,Y),covarianceofXandY,180 [a]+ =max{a,0},2 q ,weightedmeandeviation,256 IA(·),indicatorfunctionofsetA,334 sα(·),supportfunctionofsetC,337 L ((cid:2),F,P),space,399 C p dist(x,A),distancefrompointxtosetA, (cid:3)(x¯),setofLagrangemultipliersvectors, 334 348 domf,domainoffunctionf,333 N(µ,Σ),normaldistribution,16 domG,domainofmultifunctionG,365 N ,normalconetosetC,337 C R,setofextendedrealnumbers,333 (cid:5)(z),cdfofstandardnormaldistribution, epif,epigraphoffunctionf,333 16 (cid:6) ,metricprojectionontosetX,231 →e ,epiconvergence,377 X ˆ →D ,convergenceindistribution,163 SN, the set of optimal solutions of the T2(x,h),secondordertangentset,348 SAAproblem,156 X Sˆε, the set of ε-optimal solutions of the AV@R,AverageValue-at-Risk,258 N ¯ SAAproblem,181 P,setofprobabilitymeasures,306 ˆ D(A,B), deviation of set A from set B, ϑN, optimal value of the SAAproblem, 334 156 ˆ D[Z],dispersionmeasureofrandomvari- fN(x),sampleaveragefunction,155 ableZ,254 1A(·),characteristicfunctionofsetA,334 E,expectation,361 int(C),interiorofsetC,336 H(A,B),Hausdorffdistancebetweensets (cid:6)a(cid:7),integerpartofa ∈R,219 AandB,334 lscf,lowersemicontinuoushulloffunc- N,setofpositiveintegers,359 tionf,333 xi (cid:1) (cid:1) (cid:1) (cid:1) SPbook (cid:1) (cid:1) 2009/8/20 pagexii (cid:1) (cid:1) xii ListofNotations R ,radialconetosetC,337 C T ,tangentconetosetC,337 C ∇2f(x),Hessianmatrixofsecondorder partialderivatives,179 ∂,subdifferential,338 ∂◦,Clarkegeneralizedgradient,336 ∂ ,epsilonsubdifferential,380 ε posW,positivehullofmatrixW,29 Pr(A),probabilityofeventA,360 ri,relativeinterior,337 σ+,uppersemideviation,255 p σ−,lowersemideviation,255 p V@R ,Value-at-Risk,256 α Var[X],varianceofX,14 ϑ∗,optimalvalueofthetrueproblem,156 ξ[t] =(ξ1,...,ξt),historyoftheprocess, 63 a∨b=max{a,b},186 f∗,conjugateoffunctionf,338 f◦(x,d),generalizeddirectionalderiva- tive,336 g(cid:12)(x,h),directionalderivative,334 o (·),term,382 p p-efficientpoint,116 iid,independentlyidenticallydistributed, 156 (cid:1) (cid:1) (cid:1) (cid:1) SPbook (cid:1) (cid:1) 2009/8/20 pagexiii (cid:1) (cid:1) Preface Themaintopicofthisbookisoptimizationproblemsinvolvinguncertainparameters, for which stochastic models are available. Although many ways have been proposed to modeluncertainquantities, stochasticmodelshaveprovedtheirflexibilityandusefulness in diverse areas of science. This is mainly due to solid mathematical foundations and theoretical richness of the theory of probability and stochastic processes, and to sound statisticaltechniquesofusingrealdata. Optimizationproblemsinvolvingstochasticmodelsoccurinalmostallareasofscience andengineering,fromtelecommunicationandmedicinetofinance. Thisstimulatesinterest inrigorouswaysofformulating,analyzing,andsolvingsuchproblems. Duetothepresence ofrandomparametersinthemodel,thetheorycombinesconceptsoftheoptimizationtheory, thetheoryofprobabilityandstatistics,andfunctionalanalysis. Moreover,inrecentyearsthe theoryandmethodsofstochasticprogramminghaveundergonemajoradvances. Allthese factorsmotivatedustopresentinanaccessibleandrigorousformcontemporarymodelsand ideasofstochasticprogramming. Wehopethatthebookwillencourageotherresearchers toapplystochasticprogrammingmodelsandtoundertakefurtherstudiesofthisfascinating andrapidlydevelopingarea. We do not try to provide a comprehensive presentation of all aspects of stochastic programming,butweratherconcentrateontheoreticalfoundationsandrecentadvancesin selectedareas. Thebookisorganizedintosevenchapters. Thefirstchapteraddressesmod- elingissues. Thebasicconcepts,suchasrecourseactions,chance(probabilistic)constraints, and the nonanticipativity principle, are introduced in the context of specific models. The discussionisaimedatprovidingmotivationforthetheoreticaldevelopmentsinthebook, ratherthanpracticalrecommendations. Chapters2and3presentdetaileddevelopmentofthetheoryoftwo-stageandmulti- stagestochasticprogrammingproblems. Weanalyzepropertiesofthemodelsanddevelop optimality conditions and duality theory in a rather general setting. Our analysis covers generaldistributionsofuncertainparametersandprovidesspecialresultsfordiscretedistri- butions,whicharerelevantfornumericalmethods. Duetospecificpropertiesoftwo-and multistagestochasticprogrammingproblems,wewereabletoderivemanyoftheseresults withoutresortingtomethodsoffunctionalanalysis. Thebasicassumptioninthemodelingandtechnicaldevelopmentsisthattheproba- bilitydistributionoftherandomdataisnotinfluencedbyouractions(decisions). Insome applications,thisassumptioncouldbeunjustified. However,dependenceofprobabilitydis- tributionondecisionstypicallydestroystheconvexstructureoftheoptimizationproblems considered,andouranalysisexploitsconvexityinasignificantway. xiii (cid:1) (cid:1) (cid:1) (cid:1) SPbook (cid:1) (cid:1) 2009/8/20 pagexiv (cid:1) (cid:1) xiv Preface Chapter 4 deals with chance (probabilistic) constraints, which appear naturally in many applications. The chapter presents the current state of the theory, focusing on the structureoftheproblems,optimalitytheory,andduality. Wepresentgeneralizedconvexity of functions and measures, differentiability, and approximations of probability functions. Much attention is devoted to problems with separable chance constraints and problems withdiscretedistributions. Wealsoanalyzeproblemswithfirstorderstochasticdominance constraints,whichcanbeviewedasproblemswithcontinuumofprobabilisticconstraints. Manyofthepresentedresultsarerelativelynewandwerenotpreviouslyavailableinstandard textbooks. Chapter5isdevotedtostatisticalinferenceinstochasticprogramming. Thestarting point of the analysis is that the probability distribution of the random data vector is ap- proximatedbyanempiricalprobabilitymeasure. Consequently,the“true”(expectedvalue) optimizationproblemisreplacedbyitssampleaverageapproximation(SAA).Originsof thisstatisticalinferenceareintheclassicaltheoryofthemaximumlikelihoodmethodrou- tinelyusedinstatistics. Ourmotivationandapplicationsaresomewhatdifferent,because weaimatsolvingstochasticprogrammingproblemsbyMonteCarlosamplingtechniques. Thatis,thesampleisgeneratedinthecomputeranditssizeisconstrainedonlybythecom- putationalresourcesneededtosolvetheconstructedSAAproblem. Oneofthebyproducts of this theory is the complexity analysis of two-stage and multistage stochastic program- ming. Alreadyinthecaseoftwo-stagestochasticprogramming, thenumberofscenarios (discretizationpoints)growsexponentiallywithanincreaseinthenumberofrandompa- rameters. Furthermore,formultistageproblems,thecomputationalcomplexityalsogrows exponentiallywiththeincreaseofthenumberofstages. In Chapter 6 we outline the modern theory of risk averse approaches to stochastic programming. We focus on the analysis of the models, optimality theory, and duality. Staticandtwo-stageriskaversemodelsareanalyzedinmuchdetail. Wealsooutlinearisk averseapproachtomultistageproblems,usingconditionalriskmappingsandtheprinciple of“timeconsistency.” Chapter7containsformulationsoftechnicalresultsusedintheotherpartsofthebook. For some of these less-known results we give proofs, while others refer to the literature. Thesubjectindexcanhelpthereaderquicklyfindarequireddefinitionorformulationofa neededtechnicalresult. Several important aspects of stochastic programming have been left out. We do not discuss numerical methods for solving stochastic programming problems, except in section5.9,wherethestochasticapproximationmethodanditsrelationtocomplexityesti- matesareconsidered. Ofcourse,numericalmethodsisanimportanttopicwhichdeserves carefulanalysis. This,however,isavastandseparateareawhichshouldbeconsideredina moregeneralframeworkofmodernoptimizationmethodsandtoalargeextentwouldlead outsidethescopeofthisbook. Wealsodecidednottoincludeathoroughdiscussionofstochasticintegerprogram- ming. Thetheoryandmethodsofsolvingstochasticintegerprogrammingproblemsdraw heavilyfromthetheoryofgeneralintegerprogramming. Theircomprehensivepresentation wouldentaildiscussionofmanyconceptsandmethodsofthisvastfield,whichwouldhave littleconnectionwiththerestofthebook. Atthebeginningofeachchapter,weindicatetheauthorswhowereprimarilyrespon- sibleforwritingthematerial, butthebookisthecreationofallthreeofus, andweshare equalresponsibilityforerrorsandinaccuraciesthatescapedourattention. (cid:1) (cid:1) (cid:1) (cid:1)

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Optimization problems involving stochastic models occur in almost all areas of science and engineering, such as telecommunications, medicine, and finance. Their existence compels a need for rigorous ways of formulating, analyzing, and solving such problems. This book focuses on optimization problems
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