Scalarizations Of Benson Proper Efficiency In Separable Metric Space

In the framework of separable locally convex metric spaces, we give scalarization theorems on Benson proper efficiency and study Benson proper efficient solutions of vector optimization problems involving nearly subconvexlike set-valued maps.We avoid of assuming that the interior of the ordering cone is not empty and the cone has a compact base or a weakly compact base, but consider Benson proper efficiency in the framework of sparable locally convex metric spaces. First, by using polarity theorem, we give scalarization theorems on Benson proper efficiency. Applying the results to vector optimization problems with nearly subconvexlike set-valued maps, we establish scalarization theorems for Benson proper efficient solutions. Then using Hahn-Banach separation theorem, we obtain Lagrange multiplier theorem. After that, we discuss the weak duality theorem and the strong duality theorem concerning Benson efficient solutions. Finally, a new concept of Benson saddle points with respect to Lagrange set-valued maps is introduced and is used to characterize Benson proper efficient solutions. Furthermore, the convexity requirement of the objective maps is relaxed to the weakest, i.e., nearly subconvexlike set-valued maps.