Lower Bounds on the Haraux Function

TL;DR

This paper establishes new, sharper lower bounds for the Haraux function in the context of set-valued operators in Banach spaces, improving upon existing bounds even in Euclidean spaces, with applications to variational analysis.

Contribution

It introduces novel lower bounds for the Haraux function applicable to general set-valued operators, including maximally monotone operators, in reflexive Banach spaces.

Findings

New lower bounds are sharper than existing ones.

Bounds apply to Euclidean spaces, improving previous results.

Applications include bounds on the Fenchel--Young function and monotone inclusions.

Abstract

The Haraux function is an important tool in monotone operator theory and its applications. One of its salient properties for a maximally monotone operator is to be valued in $[0,+\infty]$ and to vanish only on the graph of the operator. Sharper lower bounds for this function have been proposed in specific cases. We derive lower bounds in the general context of set-valued operators in reflexive real Banach spaces. These bounds are new, even for maximally monotone operators acting on Euclidean spaces, a scenario in which we show that they can be better than existing ones. As a by-product, we obtain lower bounds on the Fenchel--Young function in variational analysis. Several examples are given and applications to composite monotone inclusions are discussed.