Optimization of Set-Valued Functions

Let X, Y, and Z be real topological vector spaces and E⊆c X be a convex set. C⊆Y, D⊆Z are to be pointed convex cones. Let F: X → 2Y be C-convex and G: X → 2Z be D-convex set-valued functions. We consider the problems [formula] This paper generalizes the Moreau-Rockafellar type theorem and the Farkas-Minkowski type theorem for set-valued functions. When Y=Rn and Z=Rm, we established the necessary and sufficient conditions for the existence of Geoffrion efficient solution of (P) and the relationship between the proper efficient solutions and Geoffrion efficient solutions of (P). The Mond-Weir type and Wolfe type vector duality theorems are also considered in this paper.