Skip to main content Accessibility help
×
Home

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

  • SHIN-YA MATSUSHITA (a1)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      ON THE CONVERGENCE RATE OF THE KRASNOSEL’SKIĬ–MANN ITERATION
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      ON THE CONVERGENCE RATE OF THE KRASNOSEL’SKIĬ–MANN ITERATION
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      ON THE CONVERGENCE RATE OF THE KRASNOSEL’SKIĬ–MANN ITERATION
      Available formats
      ×

Copyright

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.

References

Hide All
[1] Bauschke, H. H. and Combettes, P. L., Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Springer, New York, 2011).
[2] Combettes, P. L., ‘Solving monotone inclusions via compositions of nonexpansive averaged operators’, Optimization 53 (2004), 475504.
[3] 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), 757772.
[4] Corman, E. and Yuan, X., ‘A generalized proximal point algorithm and its convergence rate estimate’, SIAM J. Optim. 24 (2014), 16141638.
[5] 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.
[6] Davis, D. and Yin, W., ‘A three-operator splitting scheme and its optimization applications’, Preprint, 2015, arXiv:1504.01032.
[7] Dong, Y., ‘Comments on ‘the proximal point algorithm revisited’’, J. Optim. Theory Appl. 116 (2015), 343349.
[8] 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), 421439.
[9] 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), 293318.
[10] 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), 4176.
[11] He, B. S. and Yuan, X. M., ‘On the convergence rate of Douglas–Rachford operator splitting method’, Math. Program. 153 (2015), 715722.
[12] 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), 417429.
[13] Kamimura, S. and Takahashi, W., ‘Approximating solutions of maximal monotone operators in Hilbert spaces’, J. Approx. Theory 106 (2000), 226240.
[14] Krasnosel’skiĭ, M. A., ‘Two remarks on the method of successive approximations’, Uspekhi Mat. Nauk 10 (1955), 123127.
[15] Liang, J., Fadili, J. and Peyré, G., ‘Convergence rates with inexact nonexpansive operators’, Math. Program. 159 (2016), 403434.
[16] Lions, P. L. and Mercier, B., ‘Splitting algorithms for the sum of two nonlinear operators’, SIAM J. Numer. Anal. 16 (1979), 964979.
[17] Mann, W. R., ‘Mean value methods in iteration’, Proc. Amer. Math. Soc. 4 (1953), 506510.
[18] Martinet, B., ‘Regularisation d’inequations variationnelles par approximations successives’, Rev. Fr. Autom. Inform. Rech. Oper. 4 (1970), 154159.
[19] Matsushita, S. and Takahashi, W., ‘Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces’, Fixed Point Theory Appl. 2004 (2004), 3747.
[20] Reich, S., ‘Weak convergence theorems for nonexpansive mappings in Banach spaces’, J. Math. Anal. Appl. 67 (1979), 274276.
[21] Rockafellar, R. T., ‘Monotone operators and the proximal point algorithm’, SIAM J. Control Optim. 14 (1976), 877898.
[22] Svaiter, B. F., ‘On weak convergence of the Douglas–Rachford method’, SIAM J. Control Optim. 49 (2011), 280287.
[23] Takahashi, W., Nonlinear Functional Analysis. Fixed Point Theory and its Applications (Yokohama Publishers, Yokohama, 2000).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

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

  • SHIN-YA MATSUSHITA (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed