PDF-L.VandenbergheEE236C(Spring2013-14)Ellipsoidmethodellipsoidmethodcon

Author : liane-varnes | Published Date : 2016-12-02

EllipsoidmethodhistorydevelopedbyShorNemirovskiYudinin1970susedin1979byKhachiantoshowpolynomialsolvabilityofLPspropertieseachsteprequirescuttingplaneorsubgradientevaluationmodeststorageOn2

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L.VandenbergheEE236C(Spring2013-14)Ellipsoidmethodellipsoidmethodcon: Transcript

EllipsoidmethodhistorydevelopedbyShorNemirovskiYudinin1970susedin1979byKhachiantoshowpolynomialsolvabilityofLPspropertieseachsteprequirescuttingplaneorsubgradientevaluationmodeststorageOn2. LetAbeasquarematrixofsizenn.Thenthefollowingstatementsareequivalent.1.Aisinvertible.2.Aisrowequivalenttotheidentitymatrix.3.Ahasnpivotpositions.4.TheequationAhasonlythetrivialsolution.5.ThecolumnsofA Proximalmappingprox(x)=argminuf(u)+1 2kuxk22iffisclosedandconvexthenprox(x)existsandisuniqueforallxexistence:f(u)+(1=2)kuxk22isclosedwithboundedsublevelsetsuniqueness:f(u)+(1=2)kuxk22isstrictly Proximalmappingtheproximalmapping(prox-operator)ofaconvexfunctionhisdenedasproxh(x)=argminuh(u)+1 2kuxk22examplesh(x)=0:proxh(x)=xh(x)=IC(x)(indicatorfunctionofC):proxhisprojectiononCproxh(x)=ar First-orderconvexoptimizationmethodscomplexityofnding-suboptimalpointoffsubgradientmethod:fnondierentiablewithLipschitzconstantGO(G=)2iterationsproximalgradientmethod:f=g+h,ha`simple'nondiere AssistantAdjunctProfessor,ColumbiaUniversityCOMSW3136EssentialDataStructuresinC/C++,Spring2013,Enrollment:59COMSW3157AdvancedProgramming,Spring2013,Enrollment:136COMSW3157AdvancedProgramming,Fall2012, Self-concordantfunctionsafunctionf:Rm!Risself-concordantifdomfisanopenconvexsetfisthreetimescontinuouslydierentiableandr2f(x)0ondomffisclosed,i.e.,f(x)!1asx!bddomftheHessianoffsatisestheinequal Generalized(conic)inequalitiesconicinequality:aconstraintx2KwhereKisaconvexconeinRmwewillrequirethattheconeKisproper:closedpointed:K\(K)=f0gwithnonemptyinterior:intK=;;equivalently,K+(K)=Rmnotati Proximalpointmethoda`conceptual'algorithmforminimizingaclosedconvexfunctionf:x(k)=proxtkf(x(k1))=argminuf(u)+1 2tkkux(k1)k22canbeviewedasproximalgradientmethod(page6-3)withg(x)=0ofinterestifpro

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