Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem

Abstract

In this paper, we characterize the nonemptiness of the set of weak minimal elements for a nonempty subset of a linear space. Moreover, we obtain some existence results for a nonconvex set-valued optimization problem under weaker topological conditions. Introduction and preliminariesLet Y be a real linear space ordered by a convex cone C ⊆ Y which is assumed to be proper; i.e., {0} = C = Y . Let K be a nonempty set and F : K ⇒ Y be a set-valued mapping with nonempty values. A general form of set-valued optimization problem is usually defined as follows:(SOP) min F(x) subject to x ∈ K .There are two approaches to defining the solutions of this problem: the vector approach [3,14,17] and the set approach [15,16]. In the vector approach,x ∈ K is a solution of the problem (SOP), whenever F(x) contains a weak minimal element or minimal element of F(K ) = ∪ x∈K F(x). In the set approach, it is necessary to introduce an ordering for sets and find a minimal element of subset {F(x) : x ∈ K } of P(Y ), where P(Y ) is the set of B M. Fakhar