In this paper, we consider the Douglas-Rachford splitting method for monotone inclusion in Hilbert spaces. It can be implemented as follows: from the current iterate, first use forward-backward step to get the intermediate point, then to get the new iterate. Generally speaking, the sum operator involved in the Douglas-Rachford splitting takes the value of every intermediate point as a set. Our goal of this paper is to show that such generated set-valued sequence asymptotically includes the origin and the corresponding asymptotic inclusion speed remains desirable if the forward splitting is further Lipschitz continuous.
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View An asymptotic inclusion speed for the Douglas-Rachford splitting method in Hilbert spaces