Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity

This paper addresses the study and applications of polyhedral duality in locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar’s proper separation theorem for two convex sets one of which is polyhedral and then present its LCTV extension replacing the relative i...

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Vydáno v:	Set-valued and variational analysis Ročník 30; číslo 4; s. 1503 - 1526
Hlavní autoři:	Cuong, D. V., Mordukhovich, B. S., Nam, N. M., Sandine, G.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Dordrecht Springer Netherlands 01.12.2022 Springer Nature B.V
Témata:	Analysis Convexity Mathematics Mathematics and Statistics Optimization Geometric approach 90C31 Calculus rules Solution maps 49J53 Normal cone 49J52 Coderivative Relative interior Generalized differentiation Convex analysis
ISSN:	1877-0533, 1877-0541
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Abstract	This paper addresses the study and applications of polyhedral duality in locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar’s proper separation theorem for two convex sets one of which is polyhedral and then present its LCTV extension replacing the relative interior by its quasi-relative interior counterpart. Then we apply this result to derive enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued functions under certain polyhedrality requirements in LCTV spaces by developing a geometric approach. We also establish in this way new results on conjugate calculus and duality in convex optimization with relaxed qualification conditions in polyhedral settings. Our developments contain significant improvements to a number of existing results obtained by Ng and Song (Nonlinear Anal. 55 , 845–858, 12 ).
AbstractList	This paper addresses the study and applications of polyhedral duality in locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar’s proper separation theorem for two convex sets one of which is polyhedral and then present its LCTV extension replacing the relative interior by its quasi-relative interior counterpart. Then we apply this result to derive enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued functions under certain polyhedrality requirements in LCTV spaces by developing a geometric approach. We also establish in this way new results on conjugate calculus and duality in convex optimization with relaxed qualification conditions in polyhedral settings. Our developments contain significant improvements to a number of existing results obtained by Ng and Song (Nonlinear Anal. 55 , 845–858, 12 ). This paper addresses the study and applications of polyhedral duality in locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar’s proper separation theorem for two convex sets one of which is polyhedral and then present its LCTV extension replacing the relative interior by its quasi-relative interior counterpart. Then we apply this result to derive enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued functions under certain polyhedrality requirements in LCTV spaces by developing a geometric approach. We also establish in this way new results on conjugate calculus and duality in convex optimization with relaxed qualification conditions in polyhedral settings. Our developments contain significant improvements to a number of existing results obtained by Ng and Song (Nonlinear Anal. 55, 845–858, 12).
Author	Cuong, D. V. Mordukhovich, B. S. Sandine, G. Nam, N. M.
Author_xml	– sequence: 1 givenname: D. V. surname: Cuong fullname: Cuong, D. V. organization: Department of Mathematics, Faculty of Natural Sciences, Duy Tan University, American Degree Program, Duy Tan University – sequence: 2 givenname: B. S. surname: Mordukhovich fullname: Mordukhovich, B. S. email: boris@math.wayne.edu organization: Department of Mathematics, Wayne State University – sequence: 3 givenname: N. M. surname: Nam fullname: Nam, N. M. organization: Fariborz Maseeh Department of Mathematics and Statistics, Portland State University – sequence: 4 givenname: G. surname: Sandine fullname: Sandine, G. organization: Fariborz Maseeh Department of Mathematics and Statistics, Portland State University
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References	RockafellarRTConvex Analysis1970Princeton, NJPrinceton University Press10.1515/97814008731730193.18401 CuongDVMordukhovichBSNamNMQuasi-relative interiors for graphs of convex set-valued mappingsOptim. Lett.202115933952423829010.1007/s11590-019-01447-41471.90165 Flores-BazánFMastroeniGStrong duality in cone constrained nonconvex optimizationSIAM J. Optim.201323153169303310210.1137/1208614001285.90077 BertsekasDPNedićAOzdaglarAEConvex Analysis and Optimization2003Belmont, MAAthena Scientific1140.90001 BorweinJMLewisASPartially finite convex programming, Part I: quasi-relative interiors and duality theoryMath. Program.199257154810.1007/BF015810720778.90049 ZălinescuCA comparision of constraint qualifications in infinite-dimensional convex programing revisitedJ. Austral. Math. Soc., Sen B19994035337810.1017/S033427000001095X0926.90077 LuanNNYaoJYenNDOn some generalized polyhedral convex constructionsNumer. Funct. Anal. Optim.2017295375703763473 NgKFSongWFenchel duality in infnite-dimensional setting and its applicationsNonlinear Anal.200355845858201723110.1016/j.na.2003.07.0081045.90080 BonnansJFShapiroAPerturbation Analysis of Optimization Problems2000New YorkSpringer-Verlag10.1007/978-1-4612-1394-90966.49001 BorweinJMGoebelRNotions of relative interior in Banach spacesJ. Math. Sci.200311525422553199299110.1023/A:10229881160441136.49307 MordukhovichBSVariational Analysis and Generalized Differentiation I: Basic Theory, II: Applications2006BerlinSpringer RudinWFunctional Analysis19912nd edn.New YorkMcGraw-Hill0867.46001 HadjisavvasNSchaibleSQuasimonotone variational inequalities in Banach spacesJ. Optim. Theory Appl.19969095111139764810.1007/BF021922480904.49005 MordukhovichBSNamNMGeometric approach to convex subdifferential calculusOptimization201766839873364963110.1080/02331934.2015.11052251402.49014 Cuong, D.V., Mordukhovich, B.S., Nam, N.M., Sandine, G: Fenchel-Rockafellar theorem in infinite dimensions via generalized relative interiors. arXiv:2104.13510 (2021) ZălinescuCConvex Analysis in General Vector Spaces2002SingaporeWorld Scientific10.1142/50211023.46003 JF Bonnans (647_CR2) 2000 F Flores-Bazán (647_CR7) 2013; 23 NN Luan (647_CR9) 2017; 29 RT Rockafellar (647_CR13) 1970 DP Bertsekas (647_CR1) 2003 C Zălinescu (647_CR15) 1999; 40 BS Mordukhovich (647_CR11) 2017; 66 KF Ng (647_CR12) 2003; 55 JM Borwein (647_CR4) 1992; 57 N Hadjisavvas (647_CR8) 1996; 90 C Zălinescu (647_CR16) 2002 DV Cuong (647_CR5) 2021; 15 JM Borwein (647_CR3) 2003; 115 W Rudin (647_CR14) 1991 BS Mordukhovich (647_CR10) 2006 647_CR6
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Title	Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity
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