Skip to main content
SpringerLink
Log in

Iterative methods for solving proximal split minimization problems

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

Abstract

In this paper, we propose two iterative algorithms for finding the minimum-norm solution of a split minimization problem. We prove strong convergence of the sequences generated by the proposed algorithms. The iterative schemes are proposed in such a way that the selection of the step-sizes does not need any prior information about the operator norm. We further give some examples to numerically verify the efficiency and implementation of our new methods and compare the two algorithms presented. Our results act as supplements to several recent important results in this area.

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

Similar content being viewed by others

References

  1. Ansari, Q.H., Rehan, A.: Split Feasibility and Fixed Point Problems. In: Ansari, Q.H. (ed.) Nonlinear Analysis: Approximation Theory, Optimization and Applications, pp 281–322. Birkhäuser, Springer, New Delhi, Heidelberg, New York, Dordrecht, London (2014)

    Google Scholar 

  2. Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18(2), 441–453 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20(1), 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 221–239 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Boikanyo, O.A.: A strongly convergent algorithm for the split common fixed point problem. Appl. Math. Computat. 265, 844–853 (2015)

    Article  MathSciNet  Google Scholar 

  6. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algor. 8(2-4), 221–239 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ceng, L.-C., Ansari, Q.H., Yao, J.-C.: An extragradient method for solving split feasibility and fixed point problem. Computers Math. Appl. 64, 633–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ceng, L.-C., Ansari, Q.H., Yao, J.-C.: Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. Nonlinear Anal. 75(4), 2116–2125 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chaux, C., Combettes, P.L., Pesquet, J.-C., Wajs, V.R.: A variational formulation for frame-based inverse problems. Inverse Probl. 23, 1495–1518 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Combettes, P.L., Pesquet, J.-C.: Proximal Splitting Methods in Signal Processing. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, New York (2011)

    Chapter  Google Scholar 

  11. Kraikaew, R., Saejung, S.: On split common fixed point problems. J. Math. Anal. Appl. 415, 513–524 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lopez, G., Martin-Marquez, V., Wang, F., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl., 28. Article ID 085004 (2012)

  13. Maingé, P.E.: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Computat. Math. Appl. 59(1), 74–79 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Moudafi, A., Thakur, B.S.: Solving proximal split feasibility problems without prior knowledge of operator norms. Optim. Lett. 8, 2099–2110 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21(5), 1655–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin (1988)

    MATH  Google Scholar 

  17. Shehu, Y., Cai, G., Iyiola, O.S.: Iterative approximation of solutions for proximal split feasibility problems, Fixed Point Theory Appl., 2015. Article ID 123 (2015)

  18. Shehu, Y., Iyiola, O.S., Enyi, C.D.: An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces, Numer. Algor. 72, 835–864 (2016)

  19. Shehu, Y., Ogbuisi, F.U., Iyiola, O.S.: Convergence analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces. Optimization 15(2), 299–323 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Wang, Y., Xu, H.-K.: Strong convergence for the proximal-gradient method. J. Nonlinear Convex Anal. 15, 581–593 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Xu, H.-K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl., 26 no. 10. Article ID 105018 (2010)

  22. Xu, H.-K.: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22(6), 2021–2034 (2006)

    Article  MATH  Google Scholar 

  23. Xu, H.K.: Iterative algorithm for nonlinear operators. J. London Math. Soc. 66(2), 1–17 (2002)

    MathSciNet  Google Scholar 

  24. Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20(4), 1261–1266 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhao, J., Yang, Q.: Several solution methods for the split feasibility problem. Inverse Probl. 21(5), 1791–1799 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This research was done during the visit of first and third author at King Fahd University of Petroleum & Minerals, Dhahran Saudi Arabia. In this research, second and third author were partially supported by KFUPM Funded Research Project IN121017. The research of the fifth author was supported by the Alexander von Humboldt Foundation, Bonn, Germany. He is grateful to the Alexander von Humboldt Foundation, Bonn for the fellowship and the Institute of Mathematics, University of Wurzburg, Germany for the hospitality and facilities. Authors are grateful to the referees for their constructive suggestions and comments which improved the previous version of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Government College University, Katchery Road, Lahore, 54000, Pakistan

    M. Abbas

  2. Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia

    M. Abbas

  3. Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

    M. AlShahrani & Q. H. Ansari

  4. Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India

    Q. H. Ansari

  5. Department of Mathematical Sciences, University of Wisconsin, Milwaukee, WI, USA

    O. S. Iyiola

  6. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    Y. Shehu

Authors
  1. M. Abbas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. AlShahrani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Q. H. Ansari
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. O. S. Iyiola
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Y. Shehu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Q. H. Ansari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abbas, M., AlShahrani, M., Ansari, Q.H. et al. Iterative methods for solving proximal split minimization problems. Numer Algor 78, 193–215 (2018). https://doi.org/10.1007/s11075-017-0372-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0372-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation