New second-order optimality conditions for directional optimality of a general set-constrained optimization problem
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.
Cite
@article{arxiv.2404.17696,
title = {New second-order optimality conditions for directional optimality of a general set-constrained optimization problem},
author = {Wei Ouyang and Jane Ye and Binbin Zhang},
journal= {arXiv preprint arXiv:2404.17696},
year = {2025}
}