Weak minimizers, minimizers and variational inequalities for set-valued functions. A blooming wreath?

Abstract

Recently, necessary and sufficient conditions in terms of variational inequalities have been introduced to characterize minimizers of convex set-valued functions. Similar results have been proved for a weaker concept of minimizers and weaker variational inequalities. The implications are proved using scalarization techniques that eventually provide original problems, not fully equivalent to the set-valued counterparts. Therefore, we try, in the course of this note, to close the network among the various notions proposed. More specifically, we prove that a minimizer is always a weak minimizer, and a solution to the stronger variational inequality always also a solution to the weak variational inequality of the same type. As a special case, we obtain a complete characterization of efficiency and weak efficiency in vector optimization by set-valued variational inequalities and their scalarizations. Indeed, this might eventually prove the usefulness of the set optimization approach to renew the study of vector optimization.

Keywords:

• Set optimization
• variational Inequalities
• Dini derivative

Acknowledgements

The authors are extremely thankful to the editor and two anonymous referees who have provided several suggestions to improve the paper to its current form.

Notes

No potential conflict of interest was reported by the authors.