Henig Proper Efficiency In Vector Optimization With Nearly Subconvexlike Set-valued Maps

In this paper, we study Henig proper efficiency with respect to a base for vector optimization problem involving nearly subconvexlike set-valued maps in the framework of locally convex topological vector spaces.At first, in the framework of locally convex topological vector spaces, by using Hahn-Banach separation theorem, we establish a scalarization theorem and a Lagrange multiplier theorem on Henig proper efficient solutions with respect to a base for vector optimization problem involving nearly subconvexlike set-valued maps. Then we obtain a weak duality theorem and a strong duality theorem concerning Henig proper efficient solutions with respect to a base. Finally, a new concept of Henig saddle point with respect to a base for Lagrange set-valued maps is introduced and utilized to characterize Henig proper efficient solutions with respect to a base. When the base of the cone we considered is bounded, super efficiency is equivalent to Henig proper efficiency. Therefore we generalize the results of Aparna Mehra from normed spaces to locally convex topological vector spaces, furthermore the convexity requirement of the objective maps is weakened to nearly subconvexlike, which is the weakest convexity of set-valued maps.