Qualitative and Generalized Differentiation Properties of Optimal Value Functions with Applications to Duality
V. S. T. Long, B. S. Mordukhovich, N. M. Nam, L. White
TL;DR
The paper analyzes the perturbed optimal value function $\mu(x)=\inf\{\phi(x,y): y\in F(x)\}$ in locally convex spaces under near convexity. It develops a near-convex analytic framework, derives qualitative properties (semicontinuity, near convexity, Lipschitz stability), and provides generalized differentiation tools including $\varepsilon$-subdifferentials and Fenchel conjugates. A central result is the exact representation $\mu^*(x^*)=(\phi^*\square F^*)(x^*,0)$ under a qualification condition, enabling a duality framework for Fenchel and Lagrangian constrained optimization. The work yields strong duality results under int-near convexity and related assumptions and demonstrates applications to Lagrangian duality, advancing sensitivity analysis and dual formulations in infinite-dimensional, set-valued contexts.
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 $ε$-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.