Douglas-Rachford splitting for a Lipschitz continuous and a strongly monotone operator
Extends linear convergence guarantees for Douglas-Rachford splitting to a broader class of operators, benefiting optimization algorithm designers.
The paper proves linear convergence of the Douglas-Rachford splitting method when one operator is Lipschitz continuous and the other is strongly monotone, a setting arising in primal-dual optimization. No concrete rates are given.
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.