Nearly Convex Optimal Value Functions and Some Related Topics

TL;DR

This paper explores the properties of nearly convex functions and sets, developing duality theory and formulas for subgradients and conjugates, with applications to optimization.

Contribution

It introduces new properties of nearly convex sets and functions, and derives the near convexity of optimal value functions under qualification conditions.

Findings

Near convexity of optimal value functions established

Formulas for subgradients and Fenchel conjugates derived

Applications to duality theory in optimization

Abstract

In this paper, we introduce new properties of the relative interior calculus for nearly convex sets, functions, and set-valued mappings. These properties are important for the development of duality theory in optimization. Then we investigate optimal value functions defined by nearly convex functions and nearly convex set-valued mappings, and derive the near convexity of the optimal value function under a qualification condition. We also develop formulas for subgradients and Fenchel conjugates of this class of functions, and explore their applications to duality theory.