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

A hybrid spectral projection-based method for solving monotone nonlinear systems and signal recovery

  • Research
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we propose a hybrid spectral gradient algorithm for solving the constrained nonlinear monotone systems with an application to signal recovery. The proposed algorithm is a convex combination of the well-known spectral parameters. An efficient formula for computing the convex hybridization parameter is also proposed by tending the proposed direction to approach the generalized quasi-Newton direction. The global convergence of this algorithm is shown using the monotone and Lipschitz continuous assumptions. Finally, a numerical comparison with other related algorithms demonstrated that the suggested algorithm outperformed them regarding iterations, function evaluation, and computational time.

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

Access this article

Log in via an institution

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.

Institutional subscriptions

Algorithm 1
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Availability of Data and Materials

Data sharing does not apply to this article as no data sets were generated or analyzed during this study.

References

  1. Dai, Z., Zhou, H., Wen, F., He, S.: Efficient predictability of stock return volatility: the role of stock market implied volatility. North Am. J. Econ. Finance 52, 101174 (2020)

    Article  Google Scholar 

  2. Figueiredo, M., Nowak, R., Wright, S.J.: Gradient projection for sparse reconstruction, application to compressed sensing and other inverse problems. IEEE J-STSP 1, 586–597 (2007)

    Google Scholar 

  3. Iusem, A.N., Solodov, M.V.: Newton-type methods with generalized distances for constrained optimization. Optimization 41, 257–278 (1997)

    Article  MathSciNet  Google Scholar 

  4. Zhao, Y.B., Li, D.H.: Monotonicity of fixed point and normal mapping associated with variational inequality and its application. SIAM J. Optim. 4, 962–973 (2001)

    Article  MathSciNet  Google Scholar 

  5. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press (1970)

  6. Zhou, G., Toh, K.C.: Superlinear convergence of a Newton-type algorithm for monotone equations. J. Optimiz. Theory App. 125, 205–221 (2005)

    Article  MathSciNet  Google Scholar 

  7. Zhou, W.J., Li, D.H.: A globally convergent BFGS method for nonlinear monotone equations without any merit functions. Math. Comput. 77, 2231–2240 (2008)

    Article  MathSciNet  Google Scholar 

  8. Sabi’u, J., Shah, A., Waziri, M.Y., Ahmed, K.: Modified Hager-Zhang conjugate gradient methods via singular value analysis for solving monotone nonlinear equations with convex constraint. Int. J. Comput. Methods 18 (2020)

  9. Sabi’u, J., Shah, A., Waziri, M.Y.: A modified Hager-Zhang conjugate gradient method with optimal choices for solving monotone nonlinear equations. Int. J. Comput. Math. 1–23 (2021)

  10. Sabi’u, J., Shah, A.: An efficient three-term conjugate gradient-type algorithm for monotone nonlinear equations. RAIRO-Oper Res. 55, 1113 (2021)

    Article  MathSciNet  Google Scholar 

  11. Waziri, M.Y., Ahmed, K., Sabi’u, J., Halilu, A.S.: Enhanced Dai–Liao conjugate gradient methods for systems of monotone nonlinear equations. SeMA J. 78, 15–51 (2021)

    Article  MathSciNet  Google Scholar 

  12. Abubakar, A.B., Sabi’u, J., Kumam, P., Shah, A.: Solving nonlinear monotone operator equations via modified SR1 update. J. Appl. Math. Comput. 1–31 (2021)

  13. Waziri, M.Y., Hungu, K.A., Sabi’u, J.: Descent Perry conjugate gradient methods for systems of monotone nonlinear equations. Numer. Algorithms. 85, 763–785 (2020)

    Article  MathSciNet  Google Scholar 

  14. Waziri, M.Y., Ahmed, K., Sabi’u, J.: A Dai–Liao conjugate gradient method via modified secant equation for system of nonlinear equations. Arab. J. Math. 9, 443–457 (2020)

    Article  MathSciNet  Google Scholar 

  15. Sabi’u, J., Shah, A., Waziri, M.Y.: Two optimal Hager-Zhang conjugate gradient methods for solving monotone nonlinear equations. Appl. Numer. Math. 153, 217–233 (2020)

    Article  MathSciNet  Google Scholar 

  16. Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7, 26–33 (1997)

    Article  MathSciNet  Google Scholar 

  17. Friedlander, A., Martínez, J.M., Molina, B., Raydan, M.: Gradient method with retards and generalizations. SIAM J. Numer. Anal. 36, 275–289 (1998)

    Article  MathSciNet  Google Scholar 

  18. Fletcher, R.: On the barzilai-borwein method. In Optimization and control with applications, pp. 235–256. Springer US (2005)

  19. Dai, Y.H., Al-Baali, M., Yang, X.: A positive Barzilai–Borwein-like stepsize and an extension for symmetric linear systems. In Numerical Analysis and Optimization: NAO-III, Muscat, Oman, January, pp. 59–75. Springer International Publishing (2015)

  20. Dai, Y.H., Hager, W.W., Schittkowski, K., Zhang, H.: The cyclic Barzilai–Borwein method for unconstrained optimization. IMA J. Numer. Anal. 26, 604–627 (2006)

    Article  MathSciNet  Google Scholar 

  21. Zhou, B., Gao, L., Dai, Y.H.: Gradient methods with adaptive step-sizes. Comput. Optim. Appl. 35, 69–86 (2006)

    Article  MathSciNet  Google Scholar 

  22. Yuan, Y.X.: A new stepsize for the steepest descent method. J. Comput. Math. 149–156 (2006)

  23. Dai, Y.H., Huang, Y., Liu, X.W.: A family of spectral gradient methods for optimization. Comput. Optim. Appl. 74, 43–65 (2019)

    Article  MathSciNet  Google Scholar 

  24. Huang, Y., Dai, Y.H., Liu, X.W., Zhang, H.: On the acceleration of the Barzilai–Borwein method. Comput. Optim. Appl. 81, 717–740 (2022)

    Article  MathSciNet  Google Scholar 

  25. Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)

    Article  MathSciNet  Google Scholar 

  26. Birgin, E.G., Martínez, J.M., Raydan, M.: Spectral projected gradient methods: review and perspectives. J. Stat. Softw. 60, 1–21 (2014)

    Article  Google Scholar 

  27. Dai, Y.H., Fletcher, R.: Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. Numerische Mathematik 100, 21–47 (2005)

    Article  MathSciNet  Google Scholar 

  28. Cruz, W., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Meth. Soft. 18, 583–599 (2003)

    Article  MathSciNet  Google Scholar 

  29. Zhang, L., Zhou, W.: Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196, 478–484 (2006)

    Article  MathSciNet  Google Scholar 

  30. Wang, C., Wang, Y., Xu, C.: A projection method for a system of nonlinear monotone equations with convex constraints. Math. Methods Oper. Res. 6, 33–46 (2007)

    Article  MathSciNet  Google Scholar 

  31. Yu, Z., Lin, J., Sun, J., Xiao, Y., Liu, L., Li, Z.: Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59, 2416–2423 (2009)

    Article  MathSciNet  Google Scholar 

  32. Liu, J., Duan, Y.: Two spectral gradient projection methods for constrained equations and their linear convergence rate. J. Inequal. Appl. 2015, 1–13 (2015)

    Article  MathSciNet  Google Scholar 

  33. Zheng, L., Yang, L., Liang, Y.: A modified spectral gradient projection method for solving non-linear monotone equations with convex constraints and its application. IEEE Access. 8, 92677–92686 (2020)

    Google Scholar 

  34. Yu, Z., Li, L., Li, P.: A family of modified spectral projection methods for nonlinear monotone equations with convex constraint. Math. Probl. Eng. 2020, 1–16 (2020)

    MathSciNet  Google Scholar 

  35. Abubakar, A.B., Kumam, P., Mohammad, H.: A note on the spectral gradient projection method for nonlinear monotone equations with applications. Comput. Appl. Math. 39, 129 (2020)

    Article  MathSciNet  Google Scholar 

  36. Li, D., Wang, S., Li, Y., Wu, J.: A convergence analysis of hybrid gradient projection algorithm for constrained nonlinear equations with applications in compressed sensing. Numerical Algorithms 95, 1325–1345 (2024)

    Article  MathSciNet  Google Scholar 

  37. Martınez, J.M.: Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math. 124, 97–121 (2000)

    Article  MathSciNet  Google Scholar 

  38. Griewank, A.: Broyden updating, the good and the bad. Optimization Stories, Documenta Mathematica. Extra Volume: Optimization Stories, pp. 301–315 (2012)

  39. Martinez, J.M., Ochi, L.S.: Sobre dois métodos de Broyden. Mat. Apl. Comput. 1, 135–143 (1982)

    MathSciNet  Google Scholar 

  40. Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MathSciNet  Google Scholar 

  41. La Cruz, W.W., Martinez, J., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75, 1429–1448 (2006)

    Article  MathSciNet  Google Scholar 

  42. Halilu, A.S., Majumder, A., Waziri, M.Y., Ahmed, K.: Signal recovery with convex constrained nonlinear monotone equations through conjugate gradient hybrid approach. Math. Comput. Simul. 187, 520–539 (2021)

    Article  MathSciNet  Google Scholar 

  43. Sabi’u, J., Aremu, K.O., Althobaiti, A., Shah, A.: scaled three-term conjugate gradient methods for solving monotone equations with application. Symmetry 14, 936 (2022)

    Article  Google Scholar 

  44. Dolan, E.D., Mor\(\acute{e}\), J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

  45. Hale, E.T., Yin, W., Zhang, Y.: A fixed-point continuation method for \(l_1\) regularized minimization with applications to compressed sensing. SIAM J. Optim. 19, 1107–1130 (2008)

    Article  MathSciNet  Google Scholar 

  46. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imag. Sci. 2, 183–202 (2009)

    Article  MathSciNet  Google Scholar 

  47. Van den Berg, E., Friedlander, M.P.: Probing the pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31, 890–912 (2008)

    Article  MathSciNet  Google Scholar 

  48. Xiao, Y.H., Wang, Q.Y., Hu, Q.J.: Non-smooth equations based method for \(l_1\)-norm problems with applications to compressed sensing. Nonlinear Anal. TMA. 74, 3570–3577 (2011)

    Article  Google Scholar 

  49. Pang, J.S.: Inexact Newton methods for the nonlinear complementarity problem. Math. Program. 36, 54–71 (1986)

    Article  MathSciNet  Google Scholar 

  50. Awwal, A.M., Kumam, P., Mohammad, H., Watthayu, W., Abubakar, A.B.: A Perry-type derivative-free algorithm for solving nonlinear system of equations and minimizing \(l_1\) regularized problem. Optimization 70, 1231–1259 (2021)

    Article  MathSciNet  Google Scholar 

  51. Xiao, Y.H., Zhu, H.: A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J. Math. Anal. Appl. 405, 310–319 (2013)

    Article  MathSciNet  Google Scholar 

  52. Ibrahim, A.H., Deepho, J., Abubakar, A.B., Adamu, A.: A three-term Polak-Ribière-Polyak derivative-free method and its application to image restoration. Sci. Afr. 13, e00880 (2021)

    Google Scholar 

  53. Liu, J.K., Lia, S.J.: A projection method for convex constrained monotone nonlinear equations with applications. Comput. Math. Appl. 70, 2442–2453 (2015)

    Article  MathSciNet  Google Scholar 

  54. Abubakar, A.B., Kumam, P., Mohammad, H., Awwal, A.M.: A Barzilai-Borwein gradient projection method for sparse signal and blurred image restoration. J. Franklin Inst. 357, 7266–7285 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is sponsored by the Tertiary Education Trust Fund (TETFund) Institutional Based Research (IBR), Yusuf Maitama Sule University, Kano-Nigeria.

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Department of Mathematics, Yusuf Maitama Sule University, Kano, PMB 3220, Kano, Nigeria

    Jamilu Sabi’u & Ado Balili

  2. Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

    Mohammed Alsubhi

Authors
  1. Jamilu Sabi’u
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ado Balili
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohammed Alsubhi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

JS, AB and MA: Conceptualization, methodology, supervision, writing - review and editing, and project administration and validation. JS, AB and MA: Formal analysis and investigation. JS, MA and AB: Writing - original draft preparation.

Corresponding author

Correspondence to Jamilu Sabi’u.

Ethics declarations

Ethical Approval

Not applicable.

Consent for Publication

All the authors have agreed and given their consent for the publication of this research paper.

Conflict of Interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sabi’u, J., Balili, A. & Alsubhi, M. A hybrid spectral projection-based method for solving monotone nonlinear systems and signal recovery. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01998-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11075-024-01998-3

Keywords

Mathematics Subject Classification (2010)