Karush–Kuhn–Tucker Multiplier Rules for Efficient Solutions of Set-Valued Equilibrium Problem with Constraints

In this paper, a type of contingent derivative of a set-valued map is proposed and applied to investigate some necessary optimality conditions for weak, Henig proper, and Benson proper efficient solutions of a set-valued equilibrium problem with constraints. Under a type of Kurcyusz–Robinson–Zowe constraint qualification, our conditions are a form of Karush–Kuhn–Tucker multiplier rules. Moreover, we employ the Hölder metric subregularity to present a new necessary condition for the equilibrium problem subject to inclusion constraint. Besides the existence of the multiplier sets, their boundedness is also derived. Some examples are provided to analyze and to illustrate that our theorems are more applicable than many recently existing ones.