On maximality of quasimonotone operators

TL;DR

This paper introduces the quasimonotone polar concept for multivalued operators, explores its properties, and examines its implications for maximal quasimonotonicity and variational inequality problems.

Contribution

It defines the quasimonotone polar, extends properties of monotone polar, and analyzes maximal quasimonotonicity and its relation to variational inequalities.

Findings

Quasimonotone polar shares properties with the monotone polar.

Characterization of quasimonotone polar via normal cones.

Connections established between quasimonotonicity and variational inequalities.

Abstract

We introduce the notion of quasimonotone polar of a multivalued operator, in a similar way as the well-known monotone polar due to Martinez-Legaz and Svaiter. We first recover several properties similar to the monotone polar, including a characterization in terms of normal cones. Next, we use it to analyze certain aspects of maximal (in the sense of graph inclusion) quasimonotonicity, and its relation to the notion of maximal quasimonotonicity introduced by Aussel and Eberhard. Furthermore, we study the connections between quasimonotonicity and Minty Variational Inequality Problems.