Linear and Nonlinear Programming, 4th Edition: International Series in Operations Research & Management Science PDF
Preview Linear and Nonlinear Programming, 4th Edition: International Series in Operations Research & Management Science
International Series in Operations Research & Management Science David G. Luenberger Yinyu Ye Linear and Nonlinear Programming Fourth Edition www.it-ebooks.info International Series in Operations Research & Management Science Volume228 SeriesEditor CamilleC.Price StephenF.AustinStateUniversity,TX,USA AssociateSeriesEditor JoeZhu WorcesterPolytechnicInstitute,MA,USA FoundingSeriesEditor FrederickS.Hillier StanfordUniversity,CA,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/6161 www.it-ebooks.info www.it-ebooks.info David G. Luenberger • Yinyu Ye Linear and Nonlinear Programming Fourth Edition 123 www.it-ebooks.info DavidG.Luenberger YinyuYe DepartmentofManagementScience DepartmentofManagementScience andEngineering andEngineering StanfordUniversity StanfordUniversity Stanford,CA,USA Stanford,CA,USA ISSN0884-8289 ISSN2214-7934 (electronic) InternationalSeriesinOperationsResearch&ManagementScience ISBN978-3-319-18841-6 ISBN978-3-319-18842-3 (eBook) DOI10.1007/978-3-319-18842-3 LibraryofCongressControlNumber:2015942692 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland1973,1984(2003reprint),2008,2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternational PublishingAGSwitzerlandispartofSpringerScience+Business Media(www. springer.com) www.it-ebooks.info ToSusan,Robert,Jill,andJenna; Daisun,Fei,Tim,andKaylee www.it-ebooks.info www.it-ebooks.info Preface Thisbookisintendedasatextcoveringthecentralconceptsofpracticaloptimiza- tion techniques. It is designed for either self-study by professionals or classroom workattheundergraduateorgraduatelevelforstudentswhohaveatechnicalback- groundinengineering,mathematics,orscience.Likethefieldofoptimizationitself, whichinvolvesmanyclassicaldisciplines,thebookshouldbeusefultosystemana- lysts,operationsresearchers,numericalanalysts,managementscientists,andother specialists from the host of disciplines from which practical optimization appli- cations are drawn. The prerequisites for convenientuse of the book are relatively modest; the prime requirementbeing some familiarity with introductoryelements of linear algebra. Certain sections and developments do assume some knowledge ofmoreadvancedconceptsoflinearalgebra,suchaseigenvectoranalysis,orsome backgroundinsetsofrealnumbers,butthetextisstructuredsothatthemainstream ofthedevelopmentcanbefaithfullypursuedwithoutrelianceonthismoreadvanced backgroundmaterial. Althoughthebookcoversprimarilymaterialthatisnowfairlystandard,thisedi- tion emphasizesmethodsthat are both state-of-the-artand popular.One major in- sight is the connectionbetween the purely analytical character of an optimization problem,expressedperhapsbypropertiesofthe necessaryconditions,and the be- havior of algorithmsused to solve a problem.This was a major theme of the first editionofthisbookandthefourtheditionexpandsandfurtherillustratesthisrela- tionship. Asintheearliereditions,thematerialinthisfourtheditionisorganizedintothree separateparts.PartI isa self-containedintroductiontolinearprogramming,a key component of optimization theory. The presentation in this part is fairly conven- tional,coveringthemainelementsoftheunderlyingtheoryoflinearprogramming, manyofthemosteffectivenumericalalgorithms,andmanyofitsimportantspecial applications. Part II, which is independent of Part I, covers the theory of uncon- strainedoptimization,includingbothderivationsoftheappropriateoptimalitycon- ditionsandan introductionto basicalgorithms.Thispartofthe bookexploresthe generalpropertiesofalgorithmsanddefinesvariousnotionsofconvergence.PartIII vii www.it-ebooks.info viii Preface extends the concepts developed in the second part to constrained optimization problems.Exceptforafewisolatedsections,thispartisalsoindependentofPartI. It is possible to go directly into Parts II and III omitting Part I, and, in fact, the bookhasbeenusedinthiswayinmanyuniversities.Eachpartofthebookcontains enoughmaterialtoformthe basisofa one-quartercourse.Ineitherclassroomuse orforself-study,itisimportantnottooverlookthesuggestedexercisesattheendof eachchapter.Theselectionsgenerallyincludeexercisesofacomputationalvariety designedtotestone’sunderstandingofaparticularalgorithm,atheoreticalvariety designed to test one’s understandingof a given theoreticaldevelopment,or of the varietythatextendsthepresentationofthechaptertonewapplicationsortheoretical areas.Oneshouldattemptatleastfourorfiveexercisesfromeachchapter.Inpro- gressingthroughthebookitwouldbeunusualtoreadstraightthroughfromcover tocover.Generally,onewillwishtoskiparound.Inordertofacilitatethismode,we haveindicatedsectionsofaspecializedordigressivenaturewithanasterisk∗. NewtothiseditionisaspecialChap.6devotedtoConicLinearProgramming,a powerfulgeneralizationofLinearProgramming.Whiletheconstraintsetina nor- mal linear program is defined by a finite number of linear inequalities of finite- dimensionalvectorvariables,theconstraintsetinconiclinearprogrammingmaybe defined, for example, as a linear combination of symmetric positive semi-definite matricesofagivendimension.Indeed,manyconicstructuresarepossibleanduse- ful in a variety of applications. It must be recognized, however, that conic linear programmingisanadvancedtopic,requiringspecialstudy. Anotherimportanttopicisan acceleratedsteepestdescentmethodthatexhibits superiorconvergenceproperties,andforthisreason,hasbecomequitepopular.The proofoftheconvergencepropertyforbothstandardandacceleratedsteepestdescent methodsarepresentedinChap.8. As the field of optimization advances, addressing greater complexity, treating problemswithevermorevariables(asinBigDatasituations),rangingoverdiverse applications. The field responds yo these challenges, developing new algorithms, building effective software, and expanding overall theory. An example of a valu- able new developmentis the work on big data problems. Surprisingly,coordinate descent, with randomly selected coordinates at each step, is quite effective as ex- plainedin Chap.8.As anotherexamplesomeproblemsareformulatedso thatthe unknownscanbesplitintotwosubgroups,therearelinearconstraintsandtheobjec- tivefunctionisseparablewithrespecttothetwogroupsofvariables.Theaugmented Lagrangiancan be computedand it is naturalto use an alternatingseries method. We discuss the alternating direction method with multipliers as a dual method in Chap.14.Interestingly,thismethodisconvergentforwhenthenumberofpartition groupsistwo,butnotforfinerpartitions. We wish to thank the many students and researchers who over the years have givenuscommentsconcerningthebookandthosewhoencouragedustocarryout thisrevision. Stanford,CA,USA D.G.Luenberger Stanford,CA,USA Y.Ye January2015 www.it-ebooks.info Contents 1 Introduction................................................... 1 1.1 Optimization ............................................. 1 1.2 TypesofProblems ........................................ 2 1.3 SizeofProblems.......................................... 5 1.4 IterativeAlgorithmsandConvergence........................ 6 PartI LinearProgramming 2 BasicPropertiesofLinearPrograms ............................. 11 2.1 Introduction.............................................. 11 2.2 ExamplesofLinearProgrammingProblems................... 14 2.3 BasicSolutions ........................................... 19 2.4 TheFundamentalTheoremofLinearProgramming............. 20 2.5 RelationstoConvexity..................................... 23 2.6 Exercises ................................................ 27 3 TheSimplexMethod ........................................... 33 3.1 Pivots ................................................... 33 3.2 AdjacentExtremePoints ................................... 38 3.3 DeterminingaMinimumFeasibleSolution.................... 42 3.4 ComputationalProcedure:SimplexMethod ................... 45 3.5 FindingaBasicFeasibleSolution............................ 49 3.6 MatrixFormoftheSimplexMethod ......................... 54 3.7 SimplexMethodforTransportationProblems.................. 56 3.8 Decomposition ........................................... 68 3.9 Summary ................................................ 72 3.10 Exercises ................................................ 73 4 DualityandComplementarity ................................... 83 4.1 DualLinearPrograms ..................................... 83 4.2 TheDualityTheorem...................................... 86 ix www.it-ebooks.info
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