Linear programming: foundations and extensions

421 Pages·2014·3.9 MB·English

by  Robert J Vanderbei (auth.)

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
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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.