A Hausdorff-type distance, a directional derivative of a set-valued map and applications in set optimization

Abstract

In this paper, we follow Kuroiwa’s set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for F. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of F.

Keywords:

• Set-valued map
• directional derivative
• coderivative
• set optimization
• optimality condition

Acknowledgements

I would like to express my deepest gratitude to Professor J. Jahn for the hospitality, kindness and discussions.

Notes

No potential conflict of interest was reported by the author.

Dedicated to Professor Johannes Jahn in honor of his 65th birthday.

Additional information

Funding

The research started during my stay at the Department of Mathematics, University of Erlangen-Nuremberg, Germany, under a Georg Forster grant from the Alexander von Humboldt Foundation, and was partially supported by the Vietnam National Foundation for Science and Technology Development [grant number 101.01-2017.20].