New second-order optimality conditions for directional optimality of a general set-constrained optimization problem

TL;DR

This paper introduces new second-order optimality conditions for general set-constrained optimization problems, including nonconvex sets, focusing on directional local optimality without requiring convexity or nonemptiness of second-order tangent sets.

Contribution

It develops novel second-order necessary and sufficient conditions for directional local optimality in nonconvex set-constrained problems, extending existing theory.

Findings

New second-order optimality conditions derived

Conditions applicable to nonconvex, possibly empty second-order tangent sets

No convexity assumption needed for the second-order tangent set

Abstract

In this paper we derive new second-order optimality conditions for a very general set-constrained optimization problem where the underlying set may be nononvex. We consider local optimality in specific directions (i.e., optimal in a directional neighborhood) in pursuit of developing these new optimality conditions. First-order necessary conditions for local optimality in given directions are provided by virtue of the corresponding directional normal cones. Utilizing the classical and/or the lower generalized support function, we obtain new second-order necessary and sufficient conditions for local optimality of general nonconvex constrained optimization problem in given directions via both the corresponding asymptotic second-order tangent cone and outer second-order tangent set. Our results do not require convexity and/or nonemptyness of the outer second-order tangent set. This is an important improvement to other results in the literature since the outer second-order tangent set can be nonconvex and empty even when the set is convex.