Monotone Linear Relations: Maximality and Fitzpatrick Functions

TL;DR

This paper investigates the properties of maximal monotonicity in linear relations using Fitzpatrick functions, extending previous work and characterizing skew relations, while also resolving a problem about the convexity of graphs.

Contribution

It provides new characterizations of maximal monotonicity for linear relations and describes skew relations via Fitzpatrick functions, addressing a known open problem.

Findings

Maximal monotonicity of linear relations can be characterized using Fitzpatrick functions.

Skew linear relations are described in terms of the Fitzpatrick family.

If a maximal monotone operator has a convex graph, then the graph is affine.

Abstract

We analyze and characterize maximal monotonicity of linear relations (set-valued operators with linear graphs). An important tool in our study are Fitzpatrick functions. The results obtained partially extend work on linear and at most single-valued operators by Phelps and Simons and by Bauschke, Borwein and Wang. Furthermore, a description of skew linear relations in terms of the Fitzpatrick family is obtained. We also answer one of Simons problems by showing that if a maximal monotone operator has a convex graph, then this graph must actually be affine.