Optimization with set relations: Conjugate Duality

The aim of this paper is to develop a conjugate duality theory for convex set–valued maps. The basic idea is to understand a convex set–valued map as a function with values in the space of closed convex subsets of R. The usual inclusion of sets provides a natural ordering relation in this space. Infimum and supremum with respect to this ordering relation can be expressed with aid of union and intersection. Our main result is a strong duality assertion formulated along the lines of classical duality theorems for extended real–valued convex functions. Acknowledgments This paper is part of the author’s Ph.D. thesis, written under supervision of Christiane Tammer and Andreas H. Hamel at Martin–Luther–Universität Halle–Wittenberg. The author wishes to express his deepest graditude.