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ON THE CONVERGENCE RATE OF THE KRASNOSEL’SKIĬ–MANN ITERATION

Part of: Equations and inequalities involving nonlinear operators Nonlinear operators and their properties

Published online by Cambridge University Press:  05 January 2017

SHIN-YA MATSUSHITA
SHIN-YA MATSUSHITA*
Affiliation:
Department of Electronics and Information Systems, Akita Prefectural University, Yuri-Honjo, Akita 015-0055, Japan email matsushita@akita-pu.ac.jp
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Abstract

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The Krasnosel’skiĭ–Mann (KM) iteration is a widely used method to solve fixed point problems. This paper investigates the convergence rate for the KM iteration. We first establish a new convergence rate for the KM iteration which improves the known big-$O$ rate to little-$o$ without any other restrictions. The proof relies on the connection between the KM iteration and a useful technique on the convergence rate of summable sequences. Then we apply the result to give new results on convergence rates for the proximal point algorithm and the Douglas–Rachford method.

Keywords

Krasnosel’skiĭ–Mann iterationconvergence ratenonexpansiveproximal point algorithmDouglas–Rachford method

MSC classification

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Secondary: 47H05: Monotone operators and generalizations 47H10: Fixed-point theorems 47J25: Iterative procedures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 1 , August 2017 , pp. 162 - 170
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2017 Australian Mathematical Publishing Association Inc.

References

Bauschke, H. H. and Combettes, P. L., Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Springer, New York, 2011).CrossRefGoogle Scholar
Combettes, P. L., ‘Solving monotone inclusions via compositions of nonexpansive averaged operators’, Optimization 53 (2004), 475–504.CrossRefGoogle Scholar
Cominetti, R., Soto, J. A. and Vaisman, J., ‘On the rate of convergence of Krasnosel’skiĭ–Mann iterations and their connection with sums of Bernoullis’, Israel J. Math. 199 (2014), 757–772.CrossRefGoogle Scholar
Corman, E. and Yuan, X., ‘A generalized proximal point algorithm and its convergence rate estimate’, SIAM J. Optim. 24 (2014), 1614–1638.CrossRefGoogle Scholar
Davis, D. and Yin, W., ‘Convergence rate analysis of several splitting schemes’, in: Splitting Methods in Communication and Imaging, Science and Engineering (eds. Glowinski, R., Osher, S. and Yin, W.) (Springer, New York), to appear.Google Scholar
Davis, D. and Yin, W., ‘A three-operator splitting scheme and its optimization applications’, Preprint, 2015, arXiv:1504.01032.Google Scholar
Dong, Y., ‘Comments on ‘the proximal point algorithm revisited’’, J. Optim. Theory Appl. 116 (2015), 343–349.CrossRefGoogle Scholar
Douglas, J. and Rachford, H. H., ‘On the numerical solution of heat conduction problems in two and three space variables’, Trans. Amer. Math. Soc. 82 (1956), 421–439.CrossRefGoogle Scholar
Eckstein, J. and Bertsekas, D. P., ‘On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators’, Math. Program. 55 (1992), 293–318.CrossRefGoogle Scholar
Glowinski, R. and Marroco, A., ‘Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problémes de Dirichlet non linéaires’, Rev. Fr. Autom. Inform. Rech. Oper. 9 (1975), 41–76.Google Scholar
He, B. S. and Yuan, X. M., ‘On the convergence rate of Douglas–Rachford operator splitting method’, Math. Program. 153 (2015), 715–722.CrossRefGoogle Scholar
Kamimura, S., Kohsaka, F. and Takahashi, W., ‘Weak and strong convergence theorems for maximal monotone operators in a Banach space’, Set-Valued Anal. 12 (2004), 417–429.CrossRefGoogle Scholar
Kamimura, S. and Takahashi, W., ‘Approximating solutions of maximal monotone operators in Hilbert spaces’, J. Approx. Theory 106 (2000), 226–240.CrossRefGoogle Scholar
Krasnosel’skiĭ, M. A., ‘Two remarks on the method of successive approximations’, Uspekhi Mat. Nauk 10 (1955), 123–127.Google Scholar
Liang, J., Fadili, J. and Peyré, G., ‘Convergence rates with inexact nonexpansive operators’, Math. Program. 159 (2016), 403–434.CrossRefGoogle Scholar
Lions, P. L. and Mercier, B., ‘Splitting algorithms for the sum of two nonlinear operators’, SIAM J. Numer. Anal. 16 (1979), 964–979.CrossRefGoogle Scholar
Mann, W. R., ‘Mean value methods in iteration’, Proc. Amer. Math. Soc. 4 (1953), 506–510.CrossRefGoogle Scholar
Martinet, B., ‘Regularisation d’inequations variationnelles par approximations successives’, Rev. Fr. Autom. Inform. Rech. Oper. 4 (1970), 154–159.Google Scholar
Matsushita, S. and Takahashi, W., ‘Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces’, Fixed Point Theory Appl. 2004 (2004), 37–47.CrossRefGoogle Scholar
Reich, S., ‘Weak convergence theorems for nonexpansive mappings in Banach spaces’, J. Math. Anal. Appl. 67 (1979), 274–276.CrossRefGoogle Scholar
Rockafellar, R. T., ‘Monotone operators and the proximal point algorithm’, SIAM J. Control Optim. 14 (1976), 877–898.CrossRefGoogle Scholar
Svaiter, B. F., ‘On weak convergence of the Douglas–Rachford method’, SIAM J. Control Optim. 49 (2011), 280–287.CrossRefGoogle Scholar
Takahashi, W., Nonlinear Functional Analysis. Fixed Point Theory and its Applications (Yokohama Publishers, Yokohama, 2000).Google Scholar