Some useful set-valued maps in set optimization

Abstract

In this article, we introduce several classes of set-valued maps which can be useful in set optimization due to their applications. Exactly, we present some set-valued maps defined by scalar and vector functions and study their properties such as continuity and convexity among others. In addition, we compute their asymptotic maps which can be employed to establish coercivity and existence results in the framework of set optimization problems. Finally, we expose some possible directions for further research.

Keywords:

• Asymptotic cone and function
• asymptotic map
• set-valued maps
• set optimization

Acknowledgements

The authors are grateful to the editor and the referees for making useful comments and remarks that improved the manuscript.

Notes

No potential conflict of interest was reported by the authors.

1 The map (C, F) is not a vector-valued function to avoid incoherences.

2 Here and in the remaining parts, for simplicity, we denote it as a sequence

Additional information

Funding

This work was supported by the Conicyt, Gobierno de Chile  [project Fondecyt 1150440]; Universidad de Tarapacá [project UTA-Mayor 4733-16] (López); Universidad Nacional de Educación a Distancia [project 2011V/PUNED/0008] (Hernández).