Optimality for set functions with values in ordered vector spaces

Let (X, , ) be a finite atomless measure space,L a convex subfamily of , andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:LY beC-convex andG:LZ beD-convex set functions. Consider the following optimization problem (P): minimizeF(), subject to L andG() _D . The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.This work was partially supported by National Science Council, Taipei, Taiwan. This paper was written while the first author was visiting at the University of Iowa, 1987-88.The authors would like to express their gratitude to the two anonymous referees for their valuable comments. Also, they would like to thank Professor P. L. Yu for his encouragement and suggestions which improved the material presented here considerably.