Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity

TL;DR

This paper extends classical convex duality and differentiation results to infinite-dimensional locally convex spaces with polyhedral structures, providing new calculus rules and duality results under relaxed conditions.

Contribution

It introduces generalized duality and differentiation principles in infinite-dimensional spaces, improving upon existing results with relaxed qualification conditions.

Findings

Extended Rockafellar's separation theorem to LCTV spaces.

Derived enhanced calculus rules for convex analysis in polyhedral settings.

Established new duality and conjugate calculus results with relaxed assumptions.

Abstract

This paper addresses the study and applications of polyhedral duality of locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar's proper separation theorem for two convex sets one which is polyhedral and then present its LCTV extension with 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 in [31].