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

Final version for SIAM Journal on Optimization

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Cited by in corpus (4)

• Generalized monotone operators and their averaged resolvents
• Doubly relaxed forward-Douglas--Rachford splitting for the sum of two nonconvex and a DC function
• An adaptive splitting algorithm for the sum of three operators
• Asymptotic behavior of a nonautonomous evolution equation governed by a quasi-nonexpansive operator