Necessary and sufficient optimality conditions for constrained vector equilibrium problems using contingent hypoderivatives

In this paper, we study the Fritz John necessary and sufficient optimality conditions for weak efficient solutions of vector equilibrium problem with constraints via contingent hypoderivatives in finite-dimensional spaces. Using the stability of objective functions at a given optimal point and assumming, in addition, that the regularity condition (RC) holds, some primal and dual necessary optimality conditions for weak efficient solutions are derived. Furthermore, a dual necessary optimality condition is also established for the case of Fréchet differentiable functions. Making use of the concept of a support function on the feasible set of vector equilibrium problems with constraints, some primal and dual sufficient optimality conditions are given for the class of stable functions and Fréchet differentiable functions at a given feasible point. As an application, several necessary and sufficient optimality conditions for weak efficient solution are also obtained with the class of Hadamard differentiable functions. Examples to illustrate our results are provided as well.