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1、APPLIEDMATHEMATICSREPORTAMR00/16SEMISMOOTHMATRIXVALUEDFUNCTIONSD.SunandJ.SunAugust,2000SEMISMOOTHMATRIXVALUEDFUNCTIONS1DefengSun2SchoolofMathematicsTheUniversityofNewSouthWales,Sydney,AustraliaJieSun3FacultyofBusinessAdministrationandtheSingapore-MITAllianceNationalUnive
2、rsityofSingapore,RepublicofSingaporeNovember17,1999Abstract.Matrixvaluedfunctionsplayanimportantroleinthedevelopmentofalgorithmsforsemidef-initeprogrammingproblems.Thispaperstudiesgeneralizeddifferentialpropertiesofsuchfunctionsrelatedtononsmooth-smoothingNewtonmethods.Th
3、efirstpartofthispaperdiscussesbasicprop-ertiessuchasthegeneralizedderivative,Rademacher’stheorem,B-derivative,directionalderivative,andsemismoothness.Thesecondpartshowsthatthematrixabsolute-valuefunction,thematrixsemidefinite-projectionfunction,andthematrixprojectiveresidu
4、alfunctionarestronglysemismooth.Keywords:Matrixfunctions,Newton’smethod,nonsmoothoptimization,semidefiniteprogramming.AMSsubjectclassification:65K05,90C25,90C33.1TheresearchwaspartiallysupportedbytheAustralianResearchCouncilandgrantRP3972073ofNationalUniversityofSingapore.
5、2E-mail:sun@maths.unsw.edu.au.3Fax:(65)779-2621E-mail:jsun@nus.edu.sg.1IntroductionLetMmnandMpqbethespacesofm×nandp×qmatrices,respectively.LetMbeasubsetofMmn.Amatrixvaluedfunction(matrixfunctionforshort)isafunctionthatmapsamatrixinMtoamatrixinMpq.Weareparticularlyconce
6、rnedwiththecasethatbothMmnandMpqarereal,symmetric,block-diagonal,andofthesameblocksizes.Althoughmanyoftheresultscouldbemademoregeneral,wewillassumethatF:S(n1,···,nm)→S(n1,···,nm),whereS(n1,···,nm)isthespaceofrealsymmetricn×nblock-diagonalmatriceswithmblocksofsizesn1,···,
7、nm.Whenm=1,wesimplywriteS(n1)asSn.ThedifferentialpropertiesofFareimportantinviewoftherecentresearchonsemidefiniteprograms(SDP)anditsgeneralization,thesemidefinitecomplementarityproblems(SDCP).Forinstance,itisshown(Tseng[14])thatthesolutionofSDPandSDCPcanbereducedtosolvingam
8、atrixequationF(X)=0,whereFisacertainmatrixmeritfunction.However,inordertodevelopNewton-typemethodsforsu
