[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

Error Sensitivity for Strongly Convergent Modifications of the Proximal Point Algorithm

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The proximal point algorithm plays an important role in finding zeros of maximal monotone operators. It has however only weak convergence in the infinite-dimensional setting. In the present paper, we provide two contraction-proximal point algorithms. The strong convergence of the two algorithms is proved under two different accuracy criteria on the errors. A new technique of argument is used, and this makes sure that our conditions, which are sufficient for the strong convergence of the algorithms, are weaker than those used by several other authors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Güler, O.: On the convergence of the proximal point algorithm for convex optimization. SIAM J. Control Optim. 29, 403–419 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Progr. Ser. A 87, 189–202 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938–945 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Marino, G., Xu, H.K.: Convergence of generalized proximal point algorithm. Comm. Pure Appl. Anal. 3, 791–808 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eckstein, J.: Approximate iterations in Bregman-function-based proximal algorithms. Math. Progr. 83, 113–123 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Han, D., He, B.S.: A new accuracy criterion for approximate proximal point algorithms. J. Math. Anal. Appl. 263, 343–354 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    Article  MATH  Google Scholar 

  9. Wang, F.: A note on the regularized proximal point algorithm. J. Glob. Optim. 50, 531–535 (2011)

    Article  MATH  Google Scholar 

  10. Boikanyo, O.A., Morosanu, G.: A proximal point algorithm converging strongly for general errors. Optim. Lett. 4, 635–641 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Boikanyo, O.A., Morosanu, G.: Four parameter proximal point algorithms. Nonlinear Anal. 74, 544–555 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xu, H.K.: A regularization method for the proximal point algorithm. J. Glob. Optim. 36, 115–125 (2006)

    Article  MATH  Google Scholar 

  13. Yao, Y., Noor, M.A.: On convergence criteria of generalized proximal point algorithms. J. Comput. Appl. Math. 217, 46–55 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, F., Cui, H.: On the contraction-proximal point algorithms with multi-parameters. J. Global Optim. 54, 485–491 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226–240 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ceng, L., Wu, S., Yao, J.: New accuracy criteria for modified approximate proximal point algorithms in Hilbert spaces. Taiwan J. Math. 12, 1691–1705 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Tian, C., Wang, F.: The contraction-proximal point algorithm with square-summable errors. Fixed Point Theory and Applications 2013, 93, 10 pages (2013)

  18. Goebel, K., Kirk, W.A.: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

Download references

Acknowledgments

The authors were indebted to the anonymous referees for their helpful suggestions and comments on this manuscript. The authors were also grateful to Professor W. A. Kirk for his careful reading of the manuscript, which particularly improved the English language presentation of this manuscript. The research of Y. Wang was supported in part by National Natural Science Foundation of China with Grant No. 11271112. This work was also supported by the Scientific Research Foundation for Ph.D. of Henan Normal University (No. qd14144). The research of F.H. Wang was supported in part by the Program for Science and Technology Innovation Talents in the Universities of the Henan Province (No. 15HASTIT013).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hong-Kun Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Wang, F. & Xu, HK. Error Sensitivity for Strongly Convergent Modifications of the Proximal Point Algorithm. J Optim Theory Appl 168, 901–916 (2016). https://doi.org/10.1007/s10957-015-0835-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0835-4

Keywords

Mathematics Subject Classification

Navigation