Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints

We consider optimality conditions and duality for a general nonsmooth set-valued vector equilibrium problem with inequality constraints. We focus on the -solution, which contains most other concepts, and the firm solution, which is hardly expressed as a -solution. To face high-level nonsmoothness, we employ several notions of contingent variations as generalized derivatives. As relaxed convexity assumptions, some types of arcwise connectedness conditions are imposed. Both necessary and sufficient optimality conditions of orders are investigated for global -solutions and local firm solutions, with consequences for Henig- and Benson-proper solutions. For duality, the Wolfe and Mond-Weir schemes are dealt with, using first-order contingent variations. We discuss weak, strong, direct and converse duality. As illustrative applications, we choose three optimization-related models: a vector minimization problem with inequality constraints, a cone saddle point problem and a variational inequality. Our results are new or improve several existing ones in the literature.