Tothefirstauthor'sparents:YangCunliangandHuangLinlinAndwife:ChenYiyingAStudyonEigenvaluesofHigher-OrderTensorsandRelatedPolynomialOptimizationProblems(高阶张量特征值和相关多项式优化问题研究)YangYuningYangQingzhi(杨宇宁杨庆之)ResponsibleEditors:LiXinZhaoYanchaoCopyright©2015bySciencePressPublishedbySciencePress16DonghuangchenggenNorthStreetBeijing100717,P.R.ChinaPrintedinBeijingAllrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,withoutthepriorwrittenpermissionofthecopyrightowner.ISBN978-7-03-043765-5PrefaceInthisbook,westudytheeigenvaluesproblemsofhigherordertensorsandseveralspecificpolynomialoptimizationproblems.Theconceptsofeigenvaluesandeigenvectorsofhigherordertensorswerepre-sentedin2005byQiandLimindependently.Sincethen,moreandmorescholarshavedevotedtothisnewresearchfieldandhavemadegreatprogressinthepastdecade.ParticularlythevariouspropertiesofH-eigenvaluesofnonnegativetensorsandsolvingmethodshavebeendeveloped.Inthefirstpartofthisbookwesystem-aticallyintroducerelatedcontents.ProblemsonfindingthemaximalorminimalH-eigenvaluesandZ-eigenvaluesarespecialclassofpolynomialoptimizationprob-lems,whilepolynomialoptimizationisattractinggreatattentionduetoitsimpor-tanceandhasbeenwitnessedrapiddevelopmentinthepastyears.ItiswellknownthatingeneralapolynomialoptimizationproblemisnonconvexandNP-hard.Sopeopleattempttousesuitablerelaxedconvexoptimizationproblemtoapproximateaconcretenonconvexoptimizationproblem.Semidefiniteprogramming(SDP)isanimportantbranchofconvexoptimizationandhasbeenstudiedextensivelyintheoryaswellasinalgorithmsinthepasttwodecades.Besidesitisimportantinitselfright,SDPhasbeenfoundtobeveryusefulastheconvexapproximationofnonconvexquadraticprogrammingandsomediscreteoptimizationproblems.Inthesecondpartofthisbookwedevotetostudyingseveralspecialpolynomialop-timizationproblemsbyusingSDPrelaxationstrategy,suchasmaximalrealtensoreigenvalueproblemandtheirextendedproblems.MostofthecontentsofthisbookarecomposedofPhDdissertationofthefirstauthor,whichhavebeenpublishedinjournals.ThisworkwassupportedbytheNationalNaturalScienceFoundationofChina(GrantNo.11271206),DoctoralFundofChineseMinistryofEducation(GrantNo.20120031110024)andtheNaturalScienceFoundationofTianjin(GrantNo.12JCY-BJC31200).YangQingzhiYangYuningNov.2014ContentsChapter1Introduction.....................................................11.1Eigenvaluesproblemsofhigherordertensors............................11.2Relatedpolynomialoptimizationproblems..............................41.3Applications............................................................61.4Spectralpropertiesandalgorithms:aliteraturereview..................91.5Theorganizationofthisbook..........................................14Chapter2SpectralPropertiesofH-eigenvalueProblemsofaNonnegativeSquareTensor..................................172.1Introduction...........................................................172.2Fromnonnegativematricestononnegativetensors.....................182.3Nonnegativeirreducibletensorsandprimitivetensors..................192.4Perron-Frobeniustheoremfornonnegativetensorsandrelatedresults..222.5Geometricsimplicity...................................................292.6TheCollatz-Wielandtformula...........................
