Iteration complexity of an inexact Douglas-Rachford method and of a Douglas-Rachford-Tseng's F-B four-operator splitting method for solving monotone inclusions

TL;DR

This paper analyzes the iteration complexity of inexact Douglas-Rachford and Douglas-Rachford-Tseng's F-B splitting methods for solving monotone inclusions, providing theoretical bounds and numerical comparisons.

Contribution

It introduces and studies the iteration complexity bounds of new inexact splitting methods for two- and four-operator monotone inclusions, extending existing theoretical frameworks.

Findings

Established iteration complexity bounds in both pointwise and ergodic senses.

Demonstrated the methods' performance through numerical experiments.

Connected the algorithms to the hybrid proximal extragradient framework.

Abstract

In this paper, we propose and study the iteration complexity of an inexact Douglas-Rachford splitting (DRS) method and a Douglas-Rachford-Tseng's forward-backward (F-B) splitting method for solving two-operator and four-operator monotone inclusions, respectively. The former method (although based on a slightly different mechanism of iteration) is motivated by the recent work of J. Eckstein and W. Yao, in which an inexact DRS method is derived from a special instance of the hybrid proximal extragradient (HPE) method of Solodov and Svaiter, while the latter one combines the proposed inexact DRS method (used as an outer iteration) with a Tseng's F-B splitting type method (used as an inner iteration) for solving the corresponding subproblems. We prove iteration complexity bounds for both algorithms in the pointwise (non-ergodic) as well as in the ergodic sense by showing that they admit two different iterations: one that can be embedded into the HPE method, for which the iteration complexity is known since the work of Monteiro and Svaiter, and another one which demands a separate analysis. Finally, we perform simple numerical experiments %on three-operator and four-operator monotone inclusions to show the performance of the proposed methods when compared with other existing algorithms.