The Pseudomonotone Polar for Multivalued Operators

TL;DR

This paper introduces a new polar concept for pseudomonotonicity of multivalued operators, extending existing polar notions and exploring their properties, maximality, and links to variational inequality problems.

Contribution

It develops a pseudomonotone polar concept for multivalued operators, paralleling known polars for monotonicity and quasimonotonicity, and characterizes related maximality notions.

Findings

The pseudomonotone polar shares properties with monotone and quasimonotone polars.

Characterization of pseudomonotonicity, maximality, and pre-maximality.

Connections established between pseudomonotonicity and variational inequality problems.

Abstract

In this work, we study the pseudomonotonicity of multivalued operators from the point of view of polarity, in an analogous way as the well-known monotone polar due to Mart\'inez-Legaz and Svaiter, and the quasimonotone polar recently introduced by Bueno and Cotrina. We show that this new polar, adapted for pseudomonotonicity, possesses analogous properties to the monotone and quasimonotone polar, among which are a characterization of pseudomonotonicity, maximality and pre-maximality. Furthermore, we characterize the notion of $D$-maximal pseudomonotonicity introduced by Hadjisavvas. We conclude this work studying the connections between pseudomonotonicity and variational inequality problems.