Optimality Conditions for Efficiency in Locally Lipschitz Vector Equilibrium Problem with Constraints in Terms of Michel–Penot’s Directional Derivatives

We provide optimality conditions in terms of Michel–Penot’s directional derivatives in locally Lipschitz vector equilibrium problem with set, inequality and equality constraints. Using this derivatives, we introduce three constraint qualifications (CQ1), (CQ1-s) and (CQ2-s), and then, we establish primal and dual Karush–Kuhn–Tucker necessary optimality conditions for a local weak efficient solution and a local efficient solution. Under suitable assumptions on the MP-pseudoconvexity and strict MP-pseudoconvexity, sufficient optimality conditions, which are very near to dual Karush–Kuhn–Tucker necessary optimality conditions, are presented. Some examples to demonstrate for our findings are also provided.