Convexity and continuity of specific set-valued maps and their extremal value functions

In this paper, we study several classes of set-valued maps, which can be used in set-valued optimization and its applications, and their respective maximum and minimum value functions. The definitions of these maps are based on scalar-valued, vector-valued, and cone-valued maps. Moreover, we consider those extremal value functions which are obtained when optimizing linear functionals … Read more

Optimality Conditions for Vector Optimization with Set-Valued Maps

Based on near convexity, we introduce the concepts of nearly convexlike set-valued maps and nearly semiconvexlike set-valued maps, give some charaterizations of them, and investigate the relationships between them. Then a Farkas-Minkowski type alternative theorem is shown under the assumption of near semiconvexlikeness. By using the alternative theorem and some other lemmas, we establish necessary … Read more