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

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.