Rectangularity and paramonotonicity of maximally monotone operators

TL;DR

This paper investigates the relationship between rectangularity and paramonotonicity in maximally monotone operators, revealing their independence and conditions under which one implies the other, with implications for optimization and variational analysis.

Contribution

It provides new theoretical results clarifying the relationship between rectangularity and paramonotonicity, including independence and implications for linear relations and Hilbert space operators.

Findings

Rectangularity and paramonotonicity are independent properties.

Rectangularity implies paramonotonicity for linear relations.

Both notions are automatic for certain displacement mappings in Hilbert spaces.

Abstract

Maximally monotone operators play a key role in modern optimization and variational analysis. Two useful subclasses are rectangular (also known as star monotone) and paramonotone operators, which were introduced by Brezis and Haraux, and by Censor, Iusem and Zenios, respectively. The former class has useful range properties while the latter class is of importance for interior point methods and duality theory. Both notions are automatic for subdifferential operators and known to coincide for certain matrices; however, more precise relationships between rectangularity and paramonotonicity were not known. Our aim is to provide new results and examples concerning these notions. It is shown that rectangularity and paramonotonicity are actually independent. Moreover, for linear relations, rectangularity implies paramonotonicity but the converse implication requires additional assumptions. We also consider continuous linear monotone operators, and we point out that in Hilbert space both notions are automatic for certain displacement mappings.