In this paper, we establish a linear convergence result for the forward-backward splitting algorithm in the finding a zero of the sum of two maximal monotone operators, where one of them is set-valued strongly monotone and the other is Lipschitz continuous. We show that our convergence rate is better than Douglas--Rachford splitting algorithm's rate used by Moursi--Vandenberghe (J Optim Theory Appl 183, 179--198, 2019) under the same assumptions in most of important subcases. In addition, the forward-backward splitting requires only one resolvent while it is necessary to compute the resolvents of both operators in Douglas--Rachford splitting, which means the forward-backward splitting is more preferable. We also compare the linear convergence rate of the forward-backward splitting to the rate of Douglas--Rachford splitting acquired by Giselsson (J Fixed Point Theory Appl 19, 2241--2270, 2017) when the Lipschitz continuity of the single-valued operator is strengthened with the cocoercivity.

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View On the linear convergence of the forward-backward splitting algorithm