Douglas-Rachford splitting for a Lipschitz continuous and a strongly monotone operator

TL;DR

This paper extends the Douglas-Rachford splitting method to cases where one operator is Lipschitz continuous and the other is strongly monotone, establishing new linear convergence results relevant for primal-dual optimization methods.

Contribution

It introduces novel linear convergence results for Douglas-Rachford splitting when applied to a Lipschitz continuous and a strongly monotone operator, broadening its applicability.

Findings

Established linear convergence under new operator conditions

Extended Douglas-Rachford applicability to primal-dual methods

Provided theoretical guarantees for convergence rates

Abstract

The Douglas-Rachford method is a popular splitting technique for finding a zero of the sum of two subdifferential operators of proper closed convex functions; more generally two maximally monotone operators. Recent results concerned with linear rates of convergence of the method require additional properties of the underlying monotone operators, such as strong monotonicity and cocoercivity. In this paper, we study the case when one operator is Lipschitz continuous but not necessarily a subdifferential operator and the other operator is strongly monotone. This situation arises in optimization methods which involve primal-dual approaches. We provide new linear convergence results in this setting.