Explicitly quasiconvex set-valued optimization

The principal aim of this paper is to extend some recent results concerning the contractibility of efficient sets and the Pareto reducibility in multicriteria explicitly quasiconvex optimization problems to similar vector optimization problems involving set-valued objective maps. To this end, an appropriate notion of generalized convexity is introduced for set-valued maps taking values in a partially ordered real linear space, which naturally extends the classical concept of explicit quasiconvexity of real-valued functions. Actually, the class of so-called explicitly cone-quasiconvex set-valued maps in particular contains the cone-convex set-valued maps, and it is contained in the class of cone-quasiconvex set-valued maps.