On the maximal monotonicity of the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function

TL;DR

This paper establishes a new maximal monotonicity result for the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function, advancing understanding in Monotone Operator Theory.

Contribution

It provides a novel maximal monotonicity theorem under Rockafellar's constraint qualification for specific operator sums, using Fenchel conjugate and Fitzpatrick function techniques.

Findings

Proves maximal monotonicity of the sum under new conditions.

Utilizes Fenchel conjugate and Fitzpatrick function methods.

Addresses a key open problem in Monotone Operator Theory.

Abstract

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that Rockafellar's constraint qualification holds. In this note, we provide a new maximal monotonicity result for the sum of a maximal monotone relation and the subdifferential operator of a proper, lower semicontinuous, sublinear function. The proof relies on Rockafellar's formula for the Fenchel conjugate of the sum as well as some results on the Fitzpatrick function.