Handbook on Semidefinite, Conic and Polynomial Optimization

954 Pages·2012·8.568 MB·English

by  Miguel F. Anjos, Jean B. Lasserre (auth.), Miguel F. Anjos, Jean B. Lasserre (eds.)

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
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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