New Second-Order Optimality Conditions for Directional Optimality of a General Set-Constrained Optimization Problem
Wei Ouyang, Jane J. Ye, Binbin Zhang · SIAM Journal on Optimization · 2025
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 nonconvex. We consider local optimality in specific directions in pursuit of developing these new optimality conditions. Utilizing the classical and/or the lower generalized support function, we obtain new second-order necessary and sufficient conditions for local optimality of the 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 nonemptiness 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.