Efficiency Conditions for Nonsmooth Vector Equilibrium Problems with Constraints via Generalized Subdifferentials

This paper is devoted to the study of optimality to a nonsmooth vector equilibrium problem with set, cone and equality constraints. Using a new approach for the Karush-Kuhn-Tucker-type efficiency condition to such problem, we introduce the constraint qualification of the (CQ1) and (CQ2) types via Clarke's subdifferentials and Michel-Penot's subdifferentials. We next provide, by using these subdifferentials, some Karush-Kuhn-Tucker (KKT for short) type necessary optimality conditions for efficiency to such problem. Under suitable assumptions on the generalized quasiconvexity/the partial derivative\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial $$\end{document}-convexity, some KKT-type necessary optimality conditions become sufficient optimality conditions. Additionally, we provide some KKT-type necessary and sufficient optimality conditions for an efficient solution which does not require that the ordering cone in the objective space has a nonempty interior to those problems. Some illustrative examples are also proposed for our findings.