Optimality conditions for the Henig efficient solution of vector equilibrium problems with constraints

Abstract

The purpose of this paper is to establish optimality conditions for vector equilibrium problems with constraints. By using the separation of convex sets, we obtain the necessary and sufficient conditions for the Henig efficient solution and the superefficient solution to the vector equilibrium problem with constraints. As applications of our results, we derive some optimality conditions to the vector variational inequality problem and the vector optimization problem with constraints.

Optimality conditions for Henig efficient and superefficient solutions of vector equilibrium problems

“…The Kuhn-Tucker necessary conditions obtained here via the Michel-Penot subdifferentials can be sharper than those expressed in terms of the Clarke subdifferentials. The results obtained in this paper are more general than those obtained by Gong [3] for vector equilibrium problems with only a set constraint, and those obtained by Long-Huang-Peng [8] for vector equilibrium problems with subconvexlike functions.…”

“…It can be seen that the applied fields of vector equilibrium problems are greatly extensive. Optimality conditions for weakly efficient solutions, efficient solutions, Henig efficient solutions, globally efficient solutions and superefficient solutions of vector equilibrium problems have been studied by many authors; see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13] and references therein. There are a lot of works to dealt with the existence and optimality conditions for Henig efficient solutions and superefficient solutions of vector equilibrium problems.…”

“…Gong [3] derived optimality conditions for Henig efficient solutions and superefficient solutions of vector equilibrium problems with a set constraint. Long-Huang-Peng [8] established optimality conditions for Henig efficient solutions and superefficient solutions of vector equilibrium problems involving a cone-constraint and a set constraint with subconvexlike functions. Khanh-Tung [7] derived optimality conditions for various efficient solutions of constrained vector equilibrium problems using approximations.…”

“…Motivated by the works [3,8,11], in this paper we establish Kuhn-Tucker necessary conditions for local Henig efficient solutions and local superefficient solutions of vector equilibrium problems involving equality, inequality and set constraints with locally Lipschitz functions under the constraint qualification of Abadie type via the Michel-Penot subdifferentials. Under assumptions on generalized convexity, Kuhn-Tucker necessary conditions for Henig efficiency and superefficiency become sufficient optimality conditions.…”

Approximate weakly efficient solutions of set-valued vector equilibrium problems

“…Gong [ 2 – 4 ] obtained optimality conditions for vector equilibrium problems with constraints under the assumption of cone-convexity, and by using a nonlinear scalarization function and Ioffe subdifferentiability he derived optimality conditions for weakly efficient solutions, Henig solutions, super efficient solutions, and globally efficient solutions to nonconvex vector equilibrium problems. Long et al [ 5 ] obtained optimality conditions for Henig efficient solutions to vector equilibrium problems with functional constrains under the assumption of near cone-subconvexlikeness. Luu et al [ 7 , 8 ] established sufficient and necessary conditions for efficient solutions to vector equilibrium problems with equality and inequality constraints and obtained the Fritz John and Karush–Kuhn–Tucker necessary optimality conditions for locally efficient solutions to vector equilibrium problems with constraints and sufficient conditions under assumptions of appropriate convexities.…”