Variational inequalities characterizing weak minimality in set optimization

TL;DR

This paper introduces weak minimizers in set optimization, establishing scalarized variational inequalities as necessary and sufficient conditions for weak minimality, with applications to vector optimization in infinite-dimensional spaces.

Contribution

It develops a new framework for weak minimality in set optimization using variational inequalities, extending to infinite-dimensional vector optimization.

Findings

Necessary and sufficient conditions for weak minimality via variational inequalities.

A Minty variational principle derived as a corollary.

Application to infinite-dimensional vector optimization.

Abstract

We introduce the notion of weak minimizer in set optimization. Necessary and sufficient conditions in terms of scalarized variational inequalities of Stampacchia and Minty type, respectively, are proved. As an application, we obtain necessary and sufficient optimality conditions for weak efficiency of vector optimization in infinite dimensional spaces. A Minty variational principle in this framework is proved as a corollary of our main result.