Linear Programming Computation

739 Pages·2023·6.82 MB·English

by  Ping-Qi PAN

Ping-Qi PAN
Linear
Programming
Computation
Second Edition
Linear Programming Computation
Ping-Qi PAN
Linear Programming
Computation
Second Edition
Ping-QiPAN
DepartmentofMathematics
SoutheastUniversity
Nanjing,China
ISBN978-981-19-0146-1 ISBN978-981-19-0147-8 (eBook)
https://doi.org/10.1007/978-981-19-0147-8
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Dedicatedtomy parents,ChaoPan and
YinyunYan;mywife, MinghuaJiang;myson,
Yunpeng;mygranddaughter,Heidi
Preface to First Edition
Linear programming(LP) foundedby Dantzig (1948–1951b)mightbe one of the
most well-known and widely used mathematical tools in the world. As a branch
ofoptimization,itservesasthemostimportantcornerstoneofoperationsresearch,
decisionscience,andmanagementscience.
This branch emerged when the American mathematician George B. Dantzig
created the LP model and the simplex method in 1947. The computer, emerging
aroundthesameperiod,propelledthedevelopmentofLPandthesimplexmethod
toward practical applications. As a basic branch, LP orchestrated the birth of
a number of new branches, such as nonlinear programming, network flow and
combinatorial optimization, stochastic programming, integer programming, and
complementarytheory,etc.,andactivatedthewholefieldofoperationsresearch.
A prominent feature of LP is given by its broad applications. Closely related
to LP, many individuals have made pioneering contributions in their respective
fields.Inthefieldofeconomics,inparticular,Russian-AmericaneconomistWassily
Leontief took the 1973 Nobel Economic Prize for his epoch-makingcontribution
toquantitativeanalysisofeconomicactivities.TheacademicianL.V.Kantorovich
oftheformerSovietAcademyofScienceandAmericaneconomistProfessorT.C.
Koopmanswonthe1975NobelPrizefortheiroptimalallocationtheoryofresources
usingLP.ThesameprizewasalsogiventoProfessorsK.Arrow,P.Samuelson,H.
Simon,andL.Herwricz,severaldecadeslaterwhentheypaidcloseattentiontoLP
atthestartingdaysoftheirprofessionalcareers.
Thesimplexmethodhasalsoachievedgreatsuccessinpractice.Asitisknown,
itsapplicationstomanyfields,suchaseconomy,commerce,production,scienceand
technology, and defense and military affairs, etc, have brought about astonishing
economicandsocialbenefits.ItisrecognizedasoneofTheTenAlgorithmsinthe
TwentyCentury(IEEE2002;seeCipra(2000).
After morethan 70 years since its birth,LP is now a relativelymaturebut still
developing branch. Nevertheless, there exist great challenges. The importance of
large-scale sparse LP models is nowadaysenhanced further by globalization.The
everyday practice calls upon the research community to provide more powerful
solutiontoolsjusttokeepupwiththeever-increasingproblemsizes.Therefore,this
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viii PrefacetoFirstEdition
book does not only present fundamentalmaterials but also attempts to reflect the
stateoftheartofLP.Ithasbeenmylong-lastingbeliefthatresearch,inoperations
research/management science, in particular, should be of practical value, at least
potentially.Theauthor,therefore,focusesontheories,methods,andimplementation
techniquesthatarecloselyrelatedtoLPcomputation.
This bookconsistsof two parts. PartI mainlycoversfundamentaland conven-
tional materials, such as the geometryof the feasible region,the simplex method,
dualityprinciple,anddualsimplexmethod,implementationofthesimplexmethod,
sensitivityanalysis,parametricLP,variantsofthesimplexmethod,decomposition
method,and interior-pointmethod.Inaddition,integerlinearprogramming(ILP),
differingfromLPinnature,isalso discussednotonlybecauseILPmodelscanbe
handled by solving a sequence of LP models but also because they are so rich in
practiceandformamajorapplicationareaofLPcomputations.
Part II presents published or unpublished new results achieved by the author
himself,suchaspivotrule,dualpivotrule,simplexphase-1method,dualsimplex
phase-I method, reduced simplex method, generalized reduced simplex method,
deficient-basismethod,dualdeficient-basismethod,facemethod,dualfacemethod,
and pivotal interior-pointmethod. The last chapter contains miscellaneous topics,
suchassomespecialformsoftheLPproblem,approachestointerceptingforprimal
anddualoptimalsets,practicalpricingschemes,relaxationprinciple,localduality,
decompositionprinciple,andILPmethodbasedonthegeneralizedreducedsimplex
framework.
Tomakethematerialeasiertofollowandunderstand,thealgorithmsinthisbook
were formulated and illustrative examples were worked out wherever possible. If
thebookisusedasatextbookforupper-levelundergraduate/graduatecourse,(Part
I)maybesuitabletobeasbasiccoursematerial.
Acknowledgments
At this very moment, I deeply cherish the memory of Professor Xuchu He, my
formermentorat NanjingUniversity,arousedmy interestin optimization.I honor
the Father of linear programming and the simplex method, Professor George. B.
Dantzigfortheencouragementgivenduringthe16thInternationalSymposiumon
MathematicalProgramming,LausanneEPFL., in August1997.Iam verygrateful
toProfessorMichaelSaundersofStanfordUniversityforhisthoughtfulcomments
and suggestions given in the past. He selflessly offered the MINOS 5.51 package
(the latest version of MINOS at that time) and related materials, from which my
workandthisbookbenefitedgreatly.MINOShasbeenthebenchmarkandplatform
of our computationalexperiments. I thank Professor R. Tyrrell Rockafellar of the
UniversityofWashington,ProfessorThomasF.ColemanofCornellUniversity,and
Professor Weide Zheng of Nanjing University for their support and assistance. I
wouldalso like to thankthe followingcolleaguesfortheir supportand assistance:
Professor James V. Burke of the University of Washington, Professor Liqun
PrefacetoFirstEdition ix
Qi of Hong Kong Polytechnic University, Professor Lizhi Liao of Hong Kong
BaptistUniversity,ProfessorsLeenaSuhlandDr. AchimKobersteinofPaderborn
University,ProfessorUweSuhlofFreieUniversity,andDr.PabloGuerrero-Garcia
oftheUniversityofMalaga.
This book was supported partially by projects 10871043 and 70971136 of the
NationalNaturalScienceFoundationofChina.
Nanjing,China Ping-QiPan
December2012
Preface to Second Edition
Sinceitsbirth,linearprogramming(LP)hasachievedgreatsuccessinmanyfields,
such as economy, commerce, production, science, and technology, and brought
about amazing economic and social benefits. After 70 years of development, LP
now becomes a relatively mature branch in OR/MS. Nevertheless, the academic
community faces a major challenge of the growing demand, which needs more
powerfulandrobusttoolstodealwithlarge-scaleanddifficultproblems.
To meet the challenge, this book draws materials from a practical point of
view, focusing on theories, methods, and implementation techniques that are
closely related to LP computation. Its first edition consists of two parts. Roughly
speaking, Part I covers fundamental materials of LP, and Part II includes pub-
lished/unpublishednewresultsachievedbytheauthor.
Littleattentionwasreceivedviatheauthor’sResearchGateWebpageuntilShi,
Zhang,andZhupublishedtheirbookreviewintheEuropeanJournalofOperational
Research(June2018).Sincethen,thebookquicklyattractedconsiderableattention
fromacademiccommunitiesaroundtheworld.AsofNovember6,2022,thetopten
chapterreadsareasfollows:
Chapter Reads
1.DualityPrincipleandDualSimplexMethod(PartI) 15419
2.SimplexFeasible-PointxMethod(PartII) 7861
3.IntegerLinearProgramming(ILP)(PartI) 7340
4.ImplementationoftheSimplexMethod(PartI) 3284
5.DualSimplexPhase-IMethod(PartII) 2613
6.SimplexMethod(PartI) 2576
7.VariantsoftheSimplexMethod(PartI) 2328
8.GeometryoftheFeasibleRegion(PartI) 1951
9.PivotRule(PartII) 1844
10.SimplexPhase-IMethod(PartI) 1506
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xii PrefacetoSecondEdition
The preceding data are consistent with corresponding downloads from the
Springerwebsite.
Surprisinglyenough,thechapter”DualPrincipleandDualSimplexMethod”not
only received the maximum number of reads as a whole but also almost weekly,
evenifitisdevotedtosuchaclassicaltopic.Thismightbebecausethediscussions
arecomprehensive,comparedwithpopularliterature.
Proposedforthefirsttime,theFeasible-PointSimplexMethod,whichappeared
as the final section of the chapter ”Pivotal Interior-Point Method,” received the
second-highest reads as a whole. It achieved this in a short period of 10 weeks.
Indeed,supportedbythesolidcomputationalresults,themethoditselfmightexceed
allexpectations.
ThefinalchapterofPartI,”IntegerLinearProgramming(ILP),”usuallyreceived
thesecond-highestreadsweekly.Thishappened,Iguess,becausemanyresearchers
are interested in the newly proposed controlled-branch method and controlled-
cuttingmethod,whichhavepotentialapplicationsfordevelopingnewILPsolvers.
The author’s main contributions seem to have not been fully valued by now,
includingmostpivotrules,reducedandD-reducedsimplexmethods,deficient-basis
anddualdeficient-basismethods,faceanddualfacemethods,andthedecomposi-
tionprinciple,includedinPartII.Thisissomewhatsurprisingsincethesemethods
are supported by solid computational results, except for the newly introduced
reduced and D-reducedsimplex methods, and the decompositionprinciple. In the
author’s view, the reduced simplex method is particularly noteworthy. It is the
firstsimplexmethodthatsearchesalongtheobjective-edge.Incontrasttoexisting
simplex methods, it first determines pivot row and then pivot column, and does
sowithoutanyselectionessentially.Asadualversionofit,theD-reducedsimplex
methodsharessimilarattractivefeatures,throughdeterminingpivotcolumnfirstand
pivotrowlater.Asforthedecompositionprinciple,itallowsforsolvingarbitrarily
large-scale(evendense)LPproblems,inessence,givingaglimmeroflightonsome
othertypesofseparablelarge-scaleproblems.Timewilllikelyclarifythevalueof
thesemethodsvsotherLPsolvers.
Wearealsooptimisticaboutthefaceanddualfacemethods.Withoutexploiting
sparsity, the original methods were implemented using orthogonaltransformation
with favorable computational results reported. In the second edition, we added
twochaptersdevotedtonewmethodswithLUfactorizationforsparsecomputing.
Indeed, this is a very natural development which the author cannot help himself
frombreakinginto.
Somechangesonchaptersandsectionsweremadewithcorrectionsandimprove-
ments,andthewholesecondeditionwasorganizedintotwoparts.Roughlyspeak-
ing,PartI(Foundations)containsPartIofthefirstedition,andPartII(Advances)
includesPartII.TheSimplexFeasible-PointAlgorithmwasimproved,andremoved
from the chapter ”Pivotal Interior-Point Method” to form an independentchapter
with its new title “Simplex Interior-Point Method,” since it actually represents a
newclassofinterior-pointalgorithms,whichcanbetransformedfromthetraditional
simplexalgorithms.Thetitleoftheoriginalchapterwaschangedto“FacialInterior-
Point Method,” since the remaining algorithms represent another new class of