Conjugate Duality in Set Optimization via Nonlinear Scalarization

Two approaches are applied to the set-valued optimization problem. The following problems have been examined by Corley, Luc and their colleagues: Take the union of all objective values and then search for (weakly, properly, etc.) minimal points in this union with respect to the vector ordering. This approach is called the vector approach to set optimization. The concept shifted when the set relations were popularized by Kuroiwa–Tanaka–Ha at the end of the twentieth century. They introduced six types of set relations on the power set of topological vector space using a convex ordering cone C with nonempty interior. Therefore, this approach is called the set relation approach to set optimization. For a given vector optimization problem, several approaches are applied to construct a dual problem. A difficulty lies in the fact that the minimal point in vector optimization problem is not necessarily a singleton, though it becomes a subset of the image space in general. In this paper, we first present new definitions of set-valued conjugate map based on comparison of sets (the set relation approach) followed by introducing some types of weak duality theorems. We also show convexity and continuity properties of conjugate relations. Lastly, we present some types of strong duality theorems using nonlinear scalarizing technique for set that is generalizations of Gerstewitz’s scalarizing function for the vector-valued case.