On $\epsilon$-OPTIMALITY CONDITIONS FOR CONVEX SET-VALUED OPTIMIZATION PROBLEMS

Abstract

In this paper, $\epsilon$-subgradients for convex set-valued maps are defined. We prove an existence theorem for $\epsilon$-subgradients of convex set-valued maps. Also, we give necessary $\epsilon$- optimality conditions for an $\epsilon$-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 $\epsilon$-subgradient sum formula for two convex set-valued maps, and then give necessary and sufficient $\epsilon$-optimality conditions for the problem (CSP).