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

Abstract

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.

Keywords:

• Set optimization problem
• nonlinear scalarizing function
• Hausdorff-type distance
• Clarke generalized directional derivative

AMS Subject Classifications:

• 49J53
• 90C29

Acknowledgments

The author is grateful to the editor and the referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [11801257]