Skip to main content
SpringerLink
Log in

On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this work, based on the gradient method and the relaxed CQ algorithm introduced by López et al. (Inverse Probl. 28, 085004, 2012), we introduce two gradient-CQ algorithms for solving the split feasibility problem in the framework of Hilbert spaces. The main advantage of the proposed method is not only that the variable stepsizes depending on the information from the current iterate not the operator norm are chosen but also that the metric projection onto half-spaces, which is convenient to be calculated, is taken into account. Then both weak and strong convergence are proved under some mild conditions. Finally, numerical experiments in signal processing reveal that the proposed algorithm is effective and outruns those of Yang, López et al., Gibali et al., and others.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, London (2011)

    Book  MATH  Google Scholar 

  3. Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Censor, Y., Elfving, T.: A multiprojection algorithms using Bregman projection in a product space. Numer. Algor. 8, 221–239 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dang, Y., Gao, Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Prob. 27, 015007 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gibali, A., Liu, L.W., Tang, Y.C.: Note on the modified relaxation CQ algorithm for the split feasibility problem. Optimization Letters. 12, 1–14 (2017)

    MathSciNet  Google Scholar 

  9. He, S., Yang, C.: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl Anal. https://doi.org/10.1155/2013/942315(2013)

  10. Halpern, B.: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957–961 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  11. López, G., Martín-márquez, V., Wang, F., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28, 085004 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Suantai, S., Pholasa, N., Cholamjiak, P.: Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems. Revista de la Real Academia de Ciencias Exactas, físicas y Naturales. Serie A. Matemáticas, pp. 1–19 (2018)

  14. Wang, F.: On the convergence of CQ algorithm with variable steps for the split equality problem. Numer. Algorithms 74(3), 927–935 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, F., Xu, H.K.: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl 102085 (2010)

  16. Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Prob. 26, 105018 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yang, Q.: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Prob. 20, 1261–1266 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang, Q.: On variable-step relaxed projection algorithm for variational inequalities. J. Math. Anal. Appl. 302, 166–179 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhao, J., Zhang, Y., Yang, Q.: Modified projection methods for the split feasibility problem and multiple-sets feasibility problem. Appl. Math. Comput. 219, 1644–1653 (2012)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors wish to thank Unit of Excellence (UOE62001). Finally, the authors would like to thank the editor and reviewers for value comments to improve this manuscript.

Funding

P. Cholamjiak was supported by Thailand Research Fund and University of Phayao grant no. RSA6180084.

Author information

Authors and Affiliations

  1. School of Science, University of Phayao, Phayao, 56000, Thailand

    Suparat Kesornprom, Nattawut Pholasa & Prasit Cholamjiak

Authors
  1. Suparat Kesornprom
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nattawut Pholasa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Prasit Cholamjiak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Prasit Cholamjiak.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kesornprom, S., Pholasa, N. & Cholamjiak, P. On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem. Numer Algor 84, 997–1017 (2020). https://doi.org/10.1007/s11075-019-00790-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00790-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation