Abstract
This note provides a simple proof of a worst-case convergence rate measured by the iteration complexity for the Douglas–Rachford operator splitting method for finding a root of the sum of two maximal monotone set-valued operators. The accuracy of an iterate to the solution set is measured by the residual of a characterization of the original problem, which is different from conventional measures such as the distance to the solution set.
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Acknowledgments
The authors are grateful to three anonymous referees for their constructive suggestions which have helped us improve the presentation of this paper substantially. In particular, one referee’s valuable comments including the suggestion of using the reflection operator to address the general case where both the operators \(A\hbox { and }B\) are set-valued are highly appreciated.
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Bingsheng He was supported by the NSFC Grant 91130007, and the grant of MOE of China 20110091110004. Xiaoming Yuan was partially supported by the General Research Fund from Hong Kong Research Grants Council: 203613.
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He, B., Yuan, X. On the convergence rate of Douglas–Rachford operator splitting method. Math. Program. 153, 715–722 (2015). https://doi.org/10.1007/s10107-014-0805-x
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DOI: https://doi.org/10.1007/s10107-014-0805-x