On second-order weak sharp minima of general nonconvex set-constrained optimization problems

TL;DR

This paper develops new second-order optimality conditions for nonconvex set-constrained optimization problems, removing common convexity and regularity assumptions to better characterize weak sharp minima.

Contribution

It introduces novel second-order conditions using support functions and tangent cones that do not require convexity or strong regularity assumptions.

Findings

Eliminates the need for convexity in constraints.

Provides second-order conditions based on tangent cones.

Does not assume uniform second-order regularity.

Abstract

This paper explores local second-order weak sharp minima for a broad class of nonconvex optimization problems. We propose novel second-order optimality conditions formulated through the use of classical and lower generalized support functions. These results are based on asymptotic second-order tangent cones and outer second-order tangent sets. Specifically, our findings eliminate the necessity of assuming convexity in the constraint set and/or the outer second-order tangent set, or the nonemptiness of the outer second-order tangent set. Furthermore, unlike traditional approaches, our sufficient conditions do not rely on strong assumptions such as the uniform second-order regularity of the constraint set and the property of uniform approximation of the critical cones.