Complexity of the relaxed Peaceman-Rachford splitting method for the sum of two maximal strongly monotone operators

TL;DR

This paper analyzes the convergence properties of the relaxed Peaceman-Rachford splitting method for solving monotone inclusions involving sums of two maximal strongly monotone operators, extending previous results and identifying conditions for convergence.

Contribution

It extends convergence results for the relaxed PR splitting method using a non-Euclidean framework and establishes new convergence rate results with specific relaxation parameters.

Findings

Convergence of iterates is guaranteed within a certain relaxation parameter interval.

New pointwise and ergodic convergence rate results are established.

An example shows divergence when the relaxation parameter is outside the interval.

Abstract

This paper considers the relaxed Peaceman-Rachford (PR) splitting method for finding an approximate solution of a monotone inclusion whose underlying operator consists of the sum of two maximal strongly monotone operators. Using general results obtained in the setting of a non-Euclidean hybrid proximal extragradient framework, we extend a previous convergence result on the iterates generated by the relaxed PR splitting method, as well as establish new pointwise and ergodic convergence rate results for the method whenever an associated relaxation parameter is within a certain interval. An example is also discussed to demonstrate that the iterates may not converge when the relaxation parameter is outside this interval.