On the optimal relaxation parameters of Krasnosel'ski-Mann iteration

This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski-Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'ski-Mann iteration, with which the algorithm is proved to achieve the optimal convergence r...

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Bibliographic Details
Published in:Optimization 2021-09, Vol.70 (9), p.1959-1986
Main Authors: He, Songnian, Dong, Qiao-Li, Tian, Hanlin, Li, Xiao-Huan
Format: Article
Language:English
Subjects:
Algorithms
averaged mapping
Douglas-Rachford splitting algorithm
Fixed points (mathematics)
forward-backward splitting algorithm
Iterative methods
Krasnosel'ski-Mann iteration
nonexpansive mapping
Operators
optimal relaxation parameter
Parameters
Splitting
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Summary:This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski-Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'ski-Mann iteration, with which the algorithm is proved to achieve the optimal convergence rate. Then we present an approximation to the optimal relaxation parameter sequence since the optimal relaxation parameter sequence involves fixed points of the operators and can't be used in actual computing. Thirdly, we apply our results to the operators splitting method, such as forward-backward splitting methods and Douglas-Rachford splitting method. Finally, numerical experiments are provided to illustrate that Krasnosel'ski-Mann iteration and the relaxed projected method with our proposed relaxation parameter sequence behave better than others.
ISSN:0233-1934
1029-4945
DOI:10.1080/02331934.2020.1767101