An additive subfamily of enlargements of a maximally monotone operator

TL;DR

This paper introduces a new subfamily of additive enlargements for maximally monotone operators, inspired by Fitzpatrick's work, which includes the epsilon-subdifferential and offers a closer structural relation.

Contribution

It defines a novel subfamily of additive enlargements of maximally monotone operators, connecting them to classical epsilon-enlargements and expanding the theoretical framework.

Findings

Some enlargements are smaller than the classical epsilon-enlargement.

The epsilon-subdifferential is recovered within the new subfamily.

All enlargements in the subfamily are additive and structurally closer to epsilon-enlargements.

Abstract

We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by Svaiter. When the operator under consideration is the subdifferential of a convex lower semicontinuous proper function, we prove that some members of the subfamily are smaller than the classical $\epsilon$-subdifferential enlargement widely used in convex analysis. We also recover the epsilon-subdifferential within the subfamily. Since they are all additive, the enlargements in our subfamily can be seen as structurally closer to the $\epsilon$-subdifferential enlargement.