Adaptive Douglas-Rachford splitting algorithm for the sum of two operators

TL;DR

This paper introduces an adaptive Douglas-Rachford algorithm capable of solving problems involving one strongly monotone and one weakly monotone operator, with proven global convergence and linear rates under Lipschitz conditions.

Contribution

It develops a new adaptive splitting method that converges globally for mixed monotonicity operators, extending the applicability of Douglas-Rachford algorithms.

Findings

Algorithm converges globally to a fixed point.

Achieves global linear convergence when one operator is Lipschitz continuous.

Sharpens existing convergence results for weakly monotone operators.

Abstract

The Douglas-Rachford algorithm is a classical and powerful splitting method for minimizing the sum of two convex functions and, more generally, finding a zero of the sum of two maximally monotone operators. Although this algorithm is well understood when the involved operators are monotone or strongly monotone, the convergence theory for weakly monotone settings is far from being complete. In this paper, we propose an adaptive Douglas-Rachford splitting algorithm for the sum of two operators, one of which is strongly monotone while the other one is weakly monotone. With appropriately chosen parameters, the algorithm converges globally to a fixed point from which we derive a solution of the problem. When one operator is Lipschitz continuous, we prove global linear convergence, which sharpens recent known results.