On -optimality Conditions for Convex Set-valued Optimization Problems

In this paper, -subgradients for convex set-valued maps are defined. We prove an existence theorem for -subgradients of convex set-valued maps. Also, we give necessary optimality conditions for an -solution of a convex set-valued optimization problem (CSP). Moreover, using the single-valued function induced from the set-valued map, we obtain theorems describing the -subgradient sum formula for two convex set-valued maps, and then give necessary and sufficient -optimality conditions for the problem (CSP).