On Optimality Conditions for Mathematical Programming Problems Based on Strong Subdifferentials

TL;DR

This paper introduces refined optimality conditions for nonsmooth, nonconvex mathematical programming problems, focusing on strongly quasiconvex constraints and utilizing the strong subdifferential to analyze normal cones.

Contribution

It develops new KKT and FJ-type optimality conditions based on the strong subdifferential for specific classes of nonsmooth, nonconvex problems.

Findings

Derived normal cone expressions for supremum functions using strong subdifferentials.

Established refined optimality conditions for strongly quasiconvex constrained problems.

Illustrated theoretical results with practical examples.

Abstract

We develop refined Karush-Kuhn-Tucker (KKT) and Fritz-John (FJ)-type optimality conditions for nonsmooth, nonconvex mathematical pro\-gra\-mming problems. We pay special attention in the case that the functional constraint belongs to a specific class of generalized convex functions known as strongly quasiconvex functions. After analyzing a specialized sub\-di\-ffe\-ren\-tial, named the strong subdifferential, we compute the normal cone of the supremum function in terms of such subdifferentials, and apply this result to the mathematical programming problem. We illustrate our important results by examples.