Douglas-Rachford Splitting for the Sum of a Lipschitz Continuous and a Strongly Monotone Operator

The Douglas-Rachford method is a popular splitting technique for finding a zero of the sum of two subdifferential operators of proper, closed, and convex functions and, 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 based on primal-dual approaches. We provide new linear convergence results in this setting.