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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Bibliographic Details
Published in:Set-valued and variational analysis Vol. 30; no. 4; pp. 1503 - 1526
Main Authors: Cuong, D. V., Mordukhovich, B. S., Nam, N. M., Sandine, G.
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.12.2022
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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 ).
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ISSN:1877-0533
1877-0541
DOI:10.1007/s11228-022-00647-y