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 \(\partial \) $\partial$ -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.