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

ABSTRACT

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.

Keywords:

• Krasnosel'ski–Mann iteration
• nonexpansive mapping
• averaged mapping
• Douglas–Rachford splitting algorithm
• optimal relaxation parameter
• forward–backward splitting algorithm

AMS Subject Classifications:

• 47H09
• 47H10
• 65J15

Acknowledgments

The authors express their thanks to the reviewers, whose constructive and valuable suggestions led to high improvements in the presentation of the results.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This study was supported by Open Fund of Tianjin Key Lab for Advanced Signal Processing [No. 2017ASP-TJ03].