Treatment of set order relations by means of a nonlinear scalarization functional: a full characterization

Abstract

In this paper, we show how a nonlinear scalarization functional can be used in order to characterize several well-known set order relations and which thus plays a key role in set optimization. By means of this functional, we derive characterizations for minimal elements of set-valued optimization problems using a set approach. Our methods do not rely on any convexity assumptions on the considered sets. Furthermore, we develop a derivative-free descent method for set optimization problems without convexity assumptions to verify the usefulness of our results.

Keywords:

• Set optimization
• set order relations
• nonlinear scalarizing functional
• descent method

Acknowledgements

The authors are truly grateful to Christiane Tammer and Constantin Zălinescu for numerous valuable comments and discussions during the preparation of this manuscript. Moreover, the authors are grateful to anonymous referees for invaluable suggestions which helped improve the original manuscript.

Notes

No potential conflict of interest was reported by the authors.