An Answer to S. Simons' Question on the Maximal Monotonicity of the Sum of a Maximal Monotone Linear Operator and a Normal Cone Operator

TL;DR

This paper confirms that the sum of a maximal monotone linear relation and a normal cone operator is maximal monotone under certain conditions, resolving a long-standing open problem in Monotone Operator Theory.

Contribution

It provides an affirmative answer to Simons' question, proving the sum theorem for maximal monotone linear relations and normal cone operators.

Findings

The sum theorem holds for maximal monotone linear relations and normal cone operators.

The proof uses Rockafellar's Fenchel conjugate formula and Fitzpatrick functions.

This resolves a major open problem in Monotone Operator Theory.

Abstract

The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar's constraint qualification - that is, whether or not "the sum theorem" is true - is the most famous open problem in Monotone Operator Theory. In his 2008 monograph "From Hahn-Banach to Monotonicity", Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons' question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a normal cone operator. The proof relies on Rockafellar's formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function.