Handbook on Semidefinite, Conic and Polynomial Optimization PDF
Preview Handbook on Semidefinite, Conic and Polynomial Optimization
International Series in Operations Research & Management Science Volume 166 SeriesEditor FrederickS.Hillier StanfordUniversity,CA,USA SpecialEditorialConsultant CamilleC.Price StephenF.AustinStateUniversity,TX,USA Forfurthervolumes: http://www.springer.com/series/6161 Miguel F. Anjos • Jean B. Lasserre Editors Handbook on Semidefinite, Conic and Polynomial Optimization 123 Editors MiguelF.Anjos JeanB.Lasserre DepartmentofMathematicsandIndustrial LAAS-CNRSandInstituteofMathematics Engineering&GERAD 7AvenueduColonelRoche E´colePolytechniquedeMontre´al 31077ToulouseCedex4 Montre´al,QC,CanadaH3C3A7 France [email protected] [email protected] ISSN0884-8289 ISBN978-1-4614-0768-3 e-ISBN978-1-4614-0769-0 DOI10.1007/978-1-4614-0769-0 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2011938887 ©SpringerScience+BusinessMedia,LLC2012 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Conicoptimizationisasignificantandthrivingresearchareawithintheoptimization community. Conic optimization is the general class of problems concerned with optimizing a linear function over the intersection of an affine space and a closed convexcone.Onespecialcaseofgreatinterestisthechoiceoftheconeofpositive semidefinite matrices for which the resulting optimization problem is called a semidefiniteoptimizationproblem. Semidefiniteoptimization,orsemidefiniteprogramming(SDP),hasbeenstudied (under different names) since at least the 1940s. Its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear opti- mization were extended to solve SDP problems (and more generally, to solve convex optimization problems with efficiently computable self-concordantbarrier functions).SomeoftheearliestapplicationsofSDPthatfollowedthisdevelopment were the solution of linear matrix inequalitiesin controltheory,and the design of polynomial-timeapproximationschemesfor hard combinatorialproblemssuch as themaximum-cutproblem. This burst of activity in the 1990s led to the publication of the Handbook of SemidefiniteProgrammingintheyear2000.ThatHandbook,editedbyWolkowicz, Saigal,andVandenberghe,providedanoverviewofmuchoftheactivityinthearea. Research into semidefinite programming has continued unabated, and a new development since the beginning of the twenty-first century has been the fruit- ful interaction with algebraic geometry through the close connections between semidefinite matrices and polynomial optimization problems. This has brought about several important new results and led to an even higher level of research activity.Much of this activity can be followed on the OptimizationOnline (http:// www.optimization-online.org)andArXiv(http://arxiv.org)websites. TheobjectiveofthisHandbookonSemidefinite,ConicandPolynomialOptimiza- tion is to providethe reader with a snapshotof the state of the art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. Our intention is to provide a compendium of the researchactivitythathastakenplacesincethepublicationoftheseminalHandbook v vi Preface mentionedabove.Itisourhopethatthiswillmotivatemoreresearchers,especially doctoralstudentsandyounggraduates,to becomeinvolvedinthese thrillingareas ofoptimization. Overview oftheHandbook The Handbookbegins with a chapter presenting the basics of semidefinite, conic, and polynomial optimization. The subsequent 30 chapters are grouped into four parts:Theory,Algorithms,Software,andApplications. Theory Thisfirstpartrepresentsapproximatelyone-thirdoftheHandbook.Itcoversmany significant theoretical developments, and several chapters reflect the interactions betweenconicoptimizationandpolynomialoptimization. Algorithms This second part documents a number of different directions in which the devel- opmentofalgorithmsis taking place.It indicatesthe breadthof approachesbeing appliedtosolveconicoptimizationproblems,includingbothinterior-pointmethods andmorerecentapproaches. Software Itis a sign of thematurityof the field thatthereare nowmanysoftwarepackages to solve small- and medium-sized semidefinite optimization problems. The first chapterofthispartprovidesanoverviewofthestateoftheart,whilethesubsequent chapters document the latest developments in three commonly used software packages. Therearealsoanumberofinterfacesthatfacilitatetheuseofconicoptimization software. We have chosen not to include these in the Handbook in order to keep thefocusonthetheoreticalandalgorithmicconceptsbehindthesolvers,andthusto helpguidethereadertothemostappropriateapproachesforspecificapplications. Likeallotheraspectsofthefield,thesoftwareofferingsareinconstantevolution. As a starting point for the interested reader, we provide the URL for the soft- ware section of the Semidefinite Programming webpage maintained by Christoph Helmberg:http://www-user.tu-chemnitz.de/∼helmberg/sdp software.html. Preface vii Applications Finally,thefourthpartisconcernedwithsomeoftheapplicationareaswhereconic optimizationhasmadeasignificantimpactinrecentyears.Severaloftheseinvolve hardcombinatorialoptimizationproblemsthatcontinuetobenefitfromtheadvances intheory,algorithms,andsoftwarementionedinthepreviousparts. Acknowledgements ThisHandbookbenefitedtremendouslyfromthegeneroushelpofallthose whokindlyagreed toreferee oneormorechapters, assistingusandtheauthors tosignificantly improve the content. They were: Kurt Anstreicher, Michel Baes, Fre´de´ric Bonnans, Valentin Brimkov,SamuelBurer,BrianBorchers,Chek-BengChua,MirjamDu¨r,AlexanderEngau,Anja Fischer, MituhiroFukuda, Joa˜oGouveia,LuigiGrippo,Stefano Gualandi,ChristophHelmberg, DidierHenrion, PhilippHungerla¨nder, MichalKocvara, NathanKrislock,WilliamMartin,John Mitchell, Baback Moghaddam, Jiawang Nie, Javier Pen˜a, Veronica Piccialli, Daniel Plaumann, Mihai Putinar, Houduo Qi, Grant Schoenebeck, Frank Sottile, David Steurer, Hajime Tanaka, ThorstenTheobald,andManuelVieira. We are also grateful for the support of GERAD in the compilation of this Handbook, and particularlythehardworkanddedicationofMs.MarilyneLavoie. Finally, financial support from the Alexander von Humboldt Foundation and the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by the first editor. Montre´alandToulouse MiguelF.Anjos JeanB.Lasserre Contents 1 IntroductiontoSemidefinite,ConicandPolynomialOptimization... 1 MiguelF.AnjosandJeanB.Lasserre PartI Theory 2 TheApproachofMomentsforPolynomialEquations ................. 25 MoniqueLaurentandPhilippRostalski 3 AlgebraicDegreeinSemidefiniteandPolynomialOptimization...... 61 KristianRanestad 4 SemidefiniteRepresentationofConvexSetsandConvexHulls....... 77 J.WilliamHeltonandJiawangNie 5 ConvexHullsofAlgebraicSets ........................................... 113 Joa˜oGouveiaandRekhaThomas 6 ConvexRelaxationsandIntegralityGaps ............................... 139 EdenChlamtacandMadhurTulsiani 7 RelaxationsofCombinatorialProblemsViaAssociationSchemes ... 171 Etienne de Klerk, Fernando M. de Oliveira Filho, andDmitriiV.Pasechnik 8 CopositiveProgramming .................................................. 201 SamuelBurer 9 InvariantSemidefinitePrograms......................................... 219 ChristineBachoc,DionC.Gijswijt,AlexanderSchrijver, andFrankVallentin 10 A“Joint+Marginal”ApproachinOptimization........................ 271 JeanB.Lasserre ix x Contents 11 AnIntroductiontoFormallyRealJordanAlgebrasand TheirApplicationsinOptimization ...................................... 297 F.Alizadeh 12 ComplementarityProblemsOverSymmetricCones:A SurveyofRecentDevelopmentsinSeveralAspects .................... 339 AkikoYoshise 13 ConvexityandSemidefiniteProgramminginDimension- FreeMatrixUnknowns .................................................... 377 J.WilliamHelton,IgorKlep,andScottMcCullough 14 PositivityandOptimization:BeyondPolynomials...................... 407 JeanB.LasserreandMihaiPutinar PartII Algorithms 15 Self-RegularInterior-PointMethodsforSemidefinite Optimization ................................................................ 437 MaziarSalahiandTama´sTerlaky 16 ElementaryOptimalityConditionsforNonlinearSDPs............... 455 FlorianJarre 17 RecentProgressinInterior-PointMethods:Cutting-Plane AlgorithmsandWarmStarts ............................................. 471 AlexanderEngau 18 ExploitingSparsityin SDPRelaxationofPolynomial OptimizationProblems .................................................... 499 SunyoungKimandMasakazuKojima 19 BlockCoordinateDescentMethodsforSemidefinite Programming ............................................................... 533 ZaiwenWen,DonaldGoldfarb,andKatyaScheinberg 20 ProjectionMethodsinConicOptimization.............................. 565 DidierHenrionandJe´roˆmeMalick 21 SDPRelaxationsforNon-CommutativePolynomialOptimization... 601 MiguelNavascue´s,StefanoPironio,andAntonioAc´ın 22 SemidefiniteProgrammingandConstraintProgramming............ 635 Willem-JanvanHoeve PartIII Software 23 TheState-of-the-ArtinConicOptimizationSoftware ................. 671 HansD.Mittelmann
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