A Hausdorff-type distance, the Clarke generalized directional derivative and applications in set optimization problems

In this paper, we introduce a Hausdorff-type distance relative to an ordering cone between two sets. We obtain some properties of the Hausdorff-type distance. In particular, we give a characterization of the Hausdorff-type distance. Moreover, we introduce the Clarke generalized directional derivative for set-valued mappings by using the nonlinear scalarizing function for l-type less order relation, which is introduced by Hernández and Rodríguez-Marín [Nonconvex scalarization in set optimization with set-valued maps. J Math Anal Appl. 2007;325:1–18]. Some properties of the Clarke generalized directional derivative are given. As applications, we present necessary and sufficient optimality conditions for set optimization problems.