Linear Programming: Foundations and Extensions (International Series in Operations Research & Management Science (196)) PDF
Preview Linear Programming: Foundations and Extensions (International Series in Operations Research & Management Science (196))
International Series in Operations Research & Management Science Robert J. Vanderbei Linear Programming Foundations and Extensions Fourth Edition International Series in Operations Research & Management Science Volume 196 SeriesEditor FrederickS.Hillier StanfordUniversity,CA,USA SpecialEditorialConsultant CamilleC.Price StephenF.AustinStateUniversity,TX,USA Forfurthervolumes: http://www.springer.com/series/6161 Robert J. Vanderbei Linear Programming Foundations and Extensions Fourth Edition 123 RobertJ.Vanderbei DepartmentofOperationsResearch andFinancialEngineering PrincetonUniversity Princeton,NewJersey,USA ISSN0884-8289 ISBN978-1-4614-7629-0 ISBN978-1-4614-7630-6(eBook) DOI10.1007/978-1-4614-7630-6 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013939593 ©SpringerScience+BusinessMediaNewYork2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered andexecutedonacomputersystem, forexclusiveusebythepurchaserofthework. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) ToKrisadee, MarisaandDiana Preface This book is about constrained optimization. It begins with a thorough treat- mentoflinearprogrammingandproceedstoconvexanalysis,networkflows,integer programming, quadratic programming, and convex optimization. Along the way, dynamic programming and the linear complementarity problem are touched on as well. Thebookaimstobeafirstintroductiontothesubject. Specificexamplesand concretealgorithmsprecedemoreabstracttopics. Nevertheless,topicscoveredare developed in some depth, a large number of numerical examples are worked out indetail,andmanyrecenttopicsareincluded,mostnotablyinterior-pointmethods. Theexercisesattheendofeachchapterbothillustratethetheoryand,insomecases, extendit. Prerequisites. Thebookisdividedintofourparts. Thefirsttwopartsassume a background only in linear algebra. For the last two parts, some knowledge of multivariatecalculusisnecessary. Inparticular,thestudentshouldknowhowtouse Lagrangemultiplierstosolvesimplecalculusproblemsin2and3dimensions. Associated software. It is good to be able to solve small problems by hand, buttheproblemsoneencountersinpracticearelarge,requiringacomputerfortheir solution. Therefore,tofullyappreciatethesubject,oneneedstosolvelarge(prac- tical) problems on a computer. An important feature of this book is that it comes withsoftwareimplementingthemajoralgorithmsdescribedherein. Atthetimeof writing,softwareforthefollowingfivealgorithmsisavailable: Thetwo-phasesimplexmethodasshowninFigure6.1. • Theself-dualsimplexmethodasshowninFigure7.1. • Thepath-followingmethodasshowninFigure18.1. • Thehomogeneousself-dualmethodasshowninFigure22.1. • The long-step homogeneous self-dual method as described in Exercise • 22.4. The programs that implement these algorithms are written in C and can be easily compiled on most hardware platforms. Students/instructors are encouraged to install and compile these programs on their local hardware. Great pains have beentakentomakethesourcecodefortheseprogramsreadable(seeAppendixA). In particular, the names of the variables in the programs are consistent with the notationofthisbook. vii viii PREFACE Therearetwowaystoruntheseprograms. Thefirstistopreparetheinputin astandardcomputer-fileformat, calledMPSformat, andtoruntheprogramusing such a file as input. The advantage of this input format is that there is an archive ofproblemsstoredinthisformat,calledtheNETLIBsuite,thatonecandownload anduseimmediately(alinktotheNETLIBsuitecanbefoundatthewebsitemen- tionedbelow). But,thisformatissomewhatarchaicand,inparticular,itisnoteasy tocreatethesefilesbyhand. Therefore,theprogramscanalsoberunfromwithina problemmodelingsystemcalledAMPL.AMPLallowsonetodescribemathemat- ical programming problems using an easy to read, yet concise, algebraic notation. ToruntheprogramswithinAMPL,onesimplytellsAMPLthenameofthesolver- program before asking that a problem be solved. The text that describes AMPL, Fourer et al. (1993) makes an excellent companion to this book. It includes a dis- cussionofmanypracticallinearprogrammingproblems.Italsohaslotsofexercises tohonethemodelingskillsofthestudent. Severalinterestingcomputerprojectscanbesuggested. Hereareafewsugges- tionsregardingthesimplexcodes: Incorporate the partial pricing strategy (see Section 8.7) into the two- • phasesimplexmethodandcompareitwithfullpricing. Incorporate the steepest-edge pivot rule (see Section 8.8) into the two- • phasesimplexmethodandcompareitwiththelargest-coefficientrule. Modify the code for either variant of the simplex method so that it can • treatboundsandrangesimplicitly(seeChapter9),andcomparetheper- formancewiththeexplicittreatmentofthesuppliedcodes. Implement a “warm-start” capability so that the sensitivity analyses dis- • cussedinChapter7canbedone. Extendthesimplexcodestobeabletohandleintegerprogrammingprob- • lemsusingthebranch-and-boundmethoddescribedinChapter23. Asfortheinterior-pointcodes,onecouldtrysomeofthefollowingprojects: Modify the code for the path-following algorithm so that it implements • the affine-scaling method (see Chapter 21), and then compare the two methods. Modifythecodeforthepath-followingmethodsothatitcantreatbounds • and ranges implicitly (see Section 20.3), and compare the performance againsttheexplicittreatmentinthegivencode. Modifythecodeforthepath-followingmethodtoimplementthehigher- • ordermethoddescribedinExercise18.5. Compare. Extendthepath-followingcodetosolvequadraticprogrammingproblems • usingthealgorithmshowninFigure24.3. Furtherextendthecodesothatitcansolveconvexoptimizationproblems • usingthealgorithmshowninFigure25.2. And,perhapsthemostinterestingprojectofall: Comparethesimplexcodesagainsttheinterior-pointcodeanddecidefor • yourselfwhichalgorithmisbetteronspecificfamiliesofproblems. PREFACE ix Thesoftwareimplementingthevariousalgorithmswasdevelopedusingconsistent datastructuresandsomakingfaircomparisonsshouldbestraightforward. Thesoft- warecanbedownloadedfromthefollowingwebsite: http://www.princeton.edu/ rvdb/LPbook/ ∼ If, in the future, further codes relating to this text are developed (for example, a self-dualnetworksimplexcode),theywillbemadeavailablethroughthiswebsite. Features. Herearesomeotherfeaturesthatdistinguishthisbookfromothers: The development of the simplex method leads to Dantzig’s parametric • self-dual method. A randomized variant of this method is shown to be immunetothetravailsofdegeneracy. Thebookgivesabalancedtreatmenttoboththetraditionalsimplexmethod • and the newer interior-point methods. The notation and analysis is de- veloped to be consistent across the methods. As a result, the self-dual simplexmethodemergesasthevariantofthesimplexmethodwithmost connectionstointerior-pointmethods. Fromthebeginningandconsistentlythroughoutthebook,linearprogram- • mingproblemsareformulatedinsymmetricform. Byhighlighting sym- metry throughout, it is hoped that the reader will more fully understand andappreciatedualitytheory. Byslightlychangingtheright-handsideintheKlee–Mintyproblem,we • areabletowritedownanexplicitdictionaryforeachvertexoftheKlee– Minty problem and thereby uncover (as a homework problem) a simple, elegantargumentwhytheKlee-Mintyproblemrequires2n 1pivotsto − solve. The chapter on regression includes an analysis of the expected number • of pivots required by the self-dual variant of the simplex method. This analysisissupportedbyanempiricalstudy. Thereisanextensivetreatmentofmoderninterior-pointmethods,includ- • ingtheprimal–dualmethod, theaffine-scalingmethod, andtheself-dual path-followingmethod. Inadditiontothetraditionalapplications,whichcomemostlyfrombusi- • nessandeconomics, thebookfeaturesotherimportantapplicationssuch astheoptimaldesignoftruss-likestructuresandL1-regression. ExercisesontheWeb. Thereisalwaysaneedforfreshexercises. Hence,Ihave createdandplantomaintainagrowingarchiveofexercisesspecificallycreatedfor use in conjunction with this book. This archive is accessible from the book’s web site: http://www.princeton.edu/ rvdb/LPbook/ ∼ Theproblemsinthearchivearearrangedaccordingtothechaptersofthisbookand usenotationconsistentwiththatdevelopedherein. Advice on solving the exercises. Some problems are routine while others are fairlychallenging.Answerstosomeoftheproblemsaregivenatthebackofthebook. x PREFACE Ingeneral,theadvicegiventomebyLeonardGross(whenIwasastudent)should helpevenonthehardproblems: followyournose. Audience. Thisbookevolvedfromlecturenotesdevelopedformyintroductory graduate course in linear programming as well as my upper-level undergraduate course. A reasonable undergraduate syllabus would cover essentially all of Part 1 (Simplex Method and Duality), the first two chapters of Part 2 (Network Flows and Applications), and the first chapter of Part 4 (Integer Programming). At the graduatelevel,thesyllabusshoulddependonthepreparationofthestudents. Fora well-preparedclass,onecouldcoverthematerialinParts1and2fairlyquicklyand thenspendmoretimeonParts3(Interior-PointMethods)and4(Extensions). Dependencies. In general, Parts 2 and 3 are completely independent of each other. Bothdepend,however,onthematerialinPart1. ThefirstChapterinPart4 (IntegerProgramming)dependsonlyonmaterialfromPart1,whereastheremaining chaptersbuildonPart3material. Acknowledgments. MyinterestinlinearprogrammingwassparkedbyRobert Garfinkel when we shared an office at Bell Labs. I would like to thank him for hisconstantencouragement,advice,andsupport. Thisbookbenefitedgreatlyfrom thethoughtfulcommentsandsuggestionsofDavidBernsteinandMichaelTodd. I would also like to thank the following colleagues for their help: Ronny Ben-Tal, LeslieHall,YoshiIkura,VictorKlee,IrvinLustig,AviMandelbaum,MarcMeke- ton, Narcis Nabona, James Orlin, Andrzej Ruszczynski, and Henry Wolkowicz. I would like to thank Gary Folven at Kluwer and Fred Hillier, the series editor, for encouraging me to undertake this project. I would like to thank my students for finding many typos and occasionally more serious errors: John Gilmartin, Jacinta Warnie,StephenWoolbert,LuciaWu,andBingYang. MythankstoErhanC¸ınlar for the many times he offered advice on questions of style. I hope this book re- flectspositivelyonhisadvice. Finally,Iwouldliketoacknowledgethesupportof the National Science Foundation and the Air Force Office of Scientific Research for supporting me while writing this book. In a time of declining resources, I am especiallygratefulfortheirsupport. Princeton,NJ,USA RobertJ.Vanderbei Preface to 2nd Edition Forthe2ndedition,manynewexerciseshavebeenadded. AlsoIhaveworked hardtodeveloponlinetoolstoaidinlearningthesimplexmethodanddualitytheory. Theseonlinetoolscanbefoundonthebook’swebpage: http://www.princeton.edu/ rvdb/LPbook/ ∼ andarementionedatappropriateplacesinthetext.Besidesthelearningtools,Ihave createdseveralonlineexercises. Theseexercisesuserandomlygeneratedproblems and therefore represent a virtually unlimited collection of “routine” exercises that can be used to test basic understanding. Pointers to these online exercises are in- cludedintheexercisessectionsatappropriatepoints. Someothernotablechangesinclude: Thechapteronnetworkflowshasbeencompletelyrewritten. Hopefully, • thenewversionisanimprovementontheoriginal. Two different fonts are now used to distinguish between the set of basic • indicesandthebasismatrix. Thefirsteditionplacedgreatemphasisonthesymmetrybetweenthepri- • mal and the dual (the negative transpose property). The second edition carriesthisfurtherwithadiscussionoftherelationshipbetweenthebasic and nonbasic matrices B and N as they appear in the primal and in the dual. We show that, even though these matrices differ (they even have differentdimensions),B 1N inthedualisthenegativetransposeofthe − correspondingmatrixintheprimal. Inthechaptersdevotedtothesimplexmethodinmatrixnotation,thecol- • lectionofvariablesz ,z ,...,z ,y ,y ,...,y wasreplaced,inthefirst 1 2 n 1 2 m edition, withthesinglearrayofvariablesy ,y ,...,y . Thiscaused 1 2 n+m great confusion as the variable y in the original notation was changed i to y in the new notation. For the second edition, I have changed the n+i notationforthesinglearraytoz ,z ,...,z . 1 2 n+m Anumberoffigureshavebeenaddedtothechaptersonconvexanalysis • andonnetworkflowproblems. The algorithm refered to as the primal–dual simplex method in the first • editionhasbeenrenamedtheparametricself-dualsimplexmethodinac- cordancewithpriorstandardusage. xi
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