Qualitative and Generalized Differentiation Properties of Optimal Value Functions with Applications to Duality

TL;DR

This paper explores advanced differentiation properties of optimal value functions in optimization, providing new calculus rules, duality frameworks, and insights applicable to both convex and nonconvex problems in infinite-dimensional spaces.

Contribution

It introduces generalized differentiation properties and duality frameworks for optimal value functions, extending classical results to broader, nonconvex, and infinite-dimensional contexts.

Findings

Derived calculus rules for $\,\epsilon$-subdifferentials

Established duality frameworks for set-valued constraints

Analyzed properties like near convexity and Lipschitz continuity

Abstract

This paper investigates general and generalized differentiation properties of the optimal value function associated with perturbed optimization problems. Fundamental results on nearly convex sets and functions in infinite-dimensional spaces are then established. We proceed by analyzing general properties of the optimal value function, including its domain, epigraph, strict epigraph, near convexity, semicontinuity, and Lipschitz-type continuity in both convex and nonconvex settings. Subsequently, we derive calculus rules and representation formulas for the $\epsilon$-subdifferentials of the optimal value function and its Fenchel conjugate. We then develop a duality framework for constrained optimization problems with set-valued constraints using the Fenchel conjugate for set-valued mappings. This approach provides new perspectives on duality in generalized settings.