OPTIMIZATION OF SET-VALUED FUNCTIONS

Let X, Y, and Z be real topological vector spaces and E c X be a convex set. C subset of or equal to Y, D subset of or equal to Z are to be pointed convex cones. Let F: X --> 2(y) be C-convex and C: X --> 2(z) be D-convex set-valued functions. We consider the problems V - minimize F(x), subject to x E G(-)(-D). (P) x epsilon E This paper generalizes the Moreau-Rockafellar type theorem and the Farkas-Minkowski type theorem for set-valued functions. When Y = R(n) and Z = R(m), 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, (C) 1994 Academic Press, Inc.