Weak Minimizers, Minimizers and Variational Inequalities for set valued Functions. A blooming wreath?

TL;DR

This paper explores the relationships between various notions of minimizers and variational inequalities in convex set-valued functions, providing characterizations of efficiency and weak efficiency in vector optimization through scalarization techniques.

Contribution

It establishes that minimizers are always weak minimizers and that solutions to stronger variational inequalities imply solutions to weaker ones, unifying different concepts in set-valued optimization.

Findings

A minimizer is always a weak minimizer.

Solutions to stronger variational inequalities imply solutions to weaker ones.

Complete characterization of efficiency and weak efficiency in vector optimization.

Abstract

In the literature, necessary and sufficient conditions in terms of variational inequalities are introduced to characterize minimizers of convex set valued functions with values in a conlinear space. Similar results are 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.