A simplified proof of weak convergence in Douglas-Rachford method to a solution of the unnderlying inclusion problem

TL;DR

This paper provides a streamlined proof of weak convergence for the Douglas-Rachford method, a key algorithm in monotone operator theory, resolving a 30-year open problem with a clearer approach.

Contribution

The paper offers a simplified, more elegant proof of weak convergence for the Douglas-Rachford method in the exact case, improving upon previous complex proofs.

Findings

Proof of weak convergence is now more accessible and less technical.

Confirms the Douglas-Rachford method reliably converges in the general case.

Enhances understanding of the algorithm's theoretical foundations.

Abstract

Douglas-Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Weak convergence in this method to a solution of the underlying monotone inclusion problem in the general case remained an open problem for 30 years and was prove by the author 7 year ago. The proof presented at that occasion was cluttered with technicalities because we considered the inexact version with summable errors. The aim of this note is to present a streamlined proof of this result.