Solution to a Monotone Inclusion Problem using the Relaxed Peaceman-Rachford Splitting Method: Convergence and its Rates

TL;DR

This paper analyzes the convergence and rates of the relaxed Peaceman-Rachford splitting method for solving monotone inclusion problems with strongly monotone operators, providing new theoretical insights and numerical validation.

Contribution

It proves convergence under specific conditions for the relaxed method, addresses an open problem on convergence intervals, and establishes convergence rates with novel analysis techniques.

Findings

Convergence is proven for certain relaxation parameters.

Pointwise and R-linear convergence rates are established.

Numerical experiments validate theoretical results.

Abstract

We consider the convergence behavior using the relaxed Peaceman-Rachford splitting method to solve the monotone inclusion problem $0 \in (A + B)(u)$, where $A, B: \Re^n \rightrightarrows \Re^n$ are maximal $\beta$-strongly monotone operators, $n \geq 1$ and $\beta > 0$. Under a technical assumption, convergence of iterates using the method on the problem is proved when either $A$ or $B$ is single-valued, and the fixed relaxation parameter $\theta$ lies in the interval $(2 + \beta, 2 + \beta + \min \{ \beta, 1/\beta \})$. With this convergence result, we address an open problem that is not settled in [20] on the convergence of these iterates for $\theta \in (2 + \beta, 2 + \beta + \min \{ \beta, 1/\beta\})$. Pointwise convergence rate results and $R$-linear convergence rate results when $\theta$ lies in the interval $[2 + \beta, 2 + \beta + \min \{ \beta, 1/\beta\})$ are also provided in the paper. Our analysis to achieve these results is atypical and hence novel. Numerical experiments are conducted on the weighted Lasso minimization problem to test the validity of the assumption .