A simplified proof of weak convergence in Douglas–Rachford method

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 proved by the author 7 years ago. That proof was cluttered with technicalities because we considered the inexact version with summable errors. In this short communication we present a streamlined proof of this result.