Affine nonexpansive operators, Attouch-Th\'era duality and the Douglas-Rachford algorithm

TL;DR

This paper revisits the affine setting of the Douglas-Rachford algorithm, providing new convergence results for affine monotone relations and exploring related duality and nonexpansive mappings.

Contribution

It offers a novel convergence analysis for affine monotone operators and extends Attouch-Théra duality within the context of the Douglas-Rachford algorithm.

Findings

Convergence results for zeros of sums of maximally monotone affine relations.

Analysis of affine nonexpansive mappings and their iterative behavior.

Applications demonstrating the theoretical results.

Abstract

The Douglas-Rachford splitting algorithm was originally proposed in 1956 to solve a system of linear equations arising from the discretization of a partial differential equation. In 1979, Lions and Mercier brought forward a very powerful extension of this method suitable to solve optimization problems. In this paper, we revisit the original affine setting. We provide a powerful convergence result for finding a zero of the sum of two maximally monotone affine relations. As a by product of our analysis, we obtain results concerning the convergence of iterates of affine nonexpansive mappings as well as Attouch-Th\'era duality. Numerous examples are presented.