Second-order optimality conditions for non-convex set-constrained optimization problems

TL;DR

This paper develops new second-order optimality conditions for non-convex set-constrained optimization problems by extending classical concepts and introducing directional variants, advancing the theoretical understanding of such problems.

Contribution

It proposes two novel approaches for establishing second-order conditions in non-convex settings, extending support functions and introducing directional tangent cones.

Findings

Extended support function concept for non-convex sets

Introduced directional regular tangent cone

Established second-order optimality conditions for non-convex problems

Abstract

In this paper we study second-order optimality conditions for non-convex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well-known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper we propose two approaches for establishing second-order optimality conditions for the non-convex case. In the first approach we extend the concept of the support function so that it is applicable to general non-convex set-constrained problems, whereas in the second approach we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.