A weakly convergent fully inexact Douglas-Rachford method with relative error tolerance

TL;DR

This paper introduces fully inexact and semi-inexact variants of the Douglas-Rachford method, allowing approximate solutions within relative error tolerances, and proves their weak convergence for solving maximal monotone operator inclusion problems.

Contribution

It develops and analyzes new inexact Douglas-Rachford algorithms that accommodate approximate subproblem solutions with proven weak convergence.

Findings

Both methods generate weakly convergent sequences.

The fully inexact method allows approximate solutions within a relative error.

The semi-inexact variant solves one subproblem exactly and the other approximately.

Abstract

Douglas-Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Each of its iterations requires the sequential solution of two proximal subproblems. The aim of this work is to present a fully inexact version of Douglas-Rachford method wherein both proximal subproblems are solved approximately within a relative error tolerance. We also present a semi-inexact variant in which the first subproblem is solved exactly and the second one inexactly. We prove that both methods generate sequences weakly convergent to the solution of the underlying inclusion problem, if any.