Characterization of Set-relations in Set Optimization via Twofold Scalar Optimization Problems

The paper presents characterizations of six types of set relations for comparison of two subsets in a vector space. By transforming set relations to twofold scalar optimization problems of some (real-valued) characterizing functions (see Definition 2.1), the set-relations can be established by solving real-valued twofold optimization problems. For each of the six set-relations, some necessary conditions and sufficient conditions are established on the negativity or the nonpositivity of the associated twofold scalar problems. Several interesting examples in $\mathbb{R}^{2}$ are provided in the end to justify this approach and the results.