Autoconjugate representers for linear monotone operators

TL;DR

This paper investigates autoconjugate functions representing continuous linear monotone operators, proving their equivalence under certain conditions and constructing multiple autoconjugate representers for the identity operator.

Contribution

It demonstrates that two known autoconjugate representers coincide for continuous linear monotone operators on reflexive spaces and constructs new autoconjugate representers for the identity operator.

Findings

The two autoconjugate representers are identical for continuous linear monotone operators.

Discontinuous linear operators and subdifferential operators do not satisfy the equivalence.

An infinite family of autoconjugate representers for the identity operator on the real line is constructed.

Abstract

Monotone operators are of central importance in modern optimization and nonlinear analysis. Their study has been revolutionized lately, due to the systematic use of the Fitzpatrick function. Pioneered by Penot and Svaiter, a topic of recent interest has been the representation of maximal monotone operators by so-called autoconjugate functions. Two explicit constructions were proposed, the first by Penot and Zalinescu in 2005, and another by Bauschke and Wang in 2007. The former requires a mild constraint qualification while the latter is based on the proximal average. We show that these two autoconjugate representers must coincide for continuous linear monotone operators on reflexive spaces. The continuity and the linearity assumption are both essential as examples of discontinuous linear operators and of subdifferential operators illustrate. Furthermore, we also construct an infinite family of autoconjugate representers for the identity operator on the real line.