Skip to main content
SpringerLink
Log in

A new hybrid CGPM-based algorithm for constrained nonlinear monotone equations with applications

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

The conjugate gradient projection method (CGPM) has good theoretical properties and numerical performance for solving large-scale nonlinear monotone equations with convex constraints. In this paper, by designing a modified adaptive line search, a new hybrid CGPM-based algorithm is proposed. The search direction satisfies the sufficient descent and trust region properties which are independent of the choices of the line search. The global convergence of the algorithm is analyzed without the Lipschitz continuity. The linear convergence rate is established under some appropriate assumptions. Some preliminary numerical experiment results are reported, which show that our proposed algorithm is promising. Finally, the proposed algorithm is extended to solve the sparse signal and image restoration problems in compressed sensing.

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

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

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

    Google Scholar 

  2. Meintjes, K., Morgan, A.P.: A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22(4), 333–361 (1987)

    MathSciNet  Google Scholar 

  3. Dai, Z.F., Zhou, H.T., Wen, F.H., He, S.Y.: Efficient predictability of stock return volatility: the role of stock market implied volatility. N. Am. J. Econ. Financ. 52, 101174 (2020)

    Article  Google Scholar 

  4. Barari, M., Karimi, H.R., Razaghian, F.: Analog circuit design optimization based on evolutionary algorithms. Math. Probl. Eng. 2014, 593684 (2014)

    Article  Google Scholar 

  5. Solodov, M.V., Svaiter, B.F.: A globally convergent inexact Newton method for systems of monotone equations. In: Fukushima, M., Qi, L.Q. (eds.) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 355–369. Springer, Boston, MA (1999)

    Google Scholar 

  6. Sun, D.F., Womersley, R., Qi, H.D.: A feasible semismooth asymptotically Newton method for mixed complementarity problems. Math. Program. 94(1), 167–187 (2002)

    Article  MathSciNet  Google Scholar 

  7. Yuan, G.L., Lu, X.W., Wei, Z.X.: BFGS trust-region method for symmetric nonlinear equations. J. Comput. Appl. Math. 230(1), 44–58 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  8. Yuan, G.L., Wei, Z.X., Lu, X.W.: A BFGS trust-region method for nonlinear equations. Computing 92(4), 317–333 (2011)

    Article  MathSciNet  Google Scholar 

  9. Luo, Y.Z., Tang, G.J., Zhou, L.N.: Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method. Appl. Soft Comput. 8(2), 1068–1073 (2008)

    Article  Google Scholar 

  10. Buhmiler, S., Krejič, N., Lužanin, Z.: Practical quasi-Newton algorithms for singular nonlinear systems. Numer. Algorithms. 55(4), 481–502 (2010)

    Article  MathSciNet  Google Scholar 

  11. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control. Optim. 37(3), 765–776 (1999)

    Article  MathSciNet  Google Scholar 

  12. Ou, Y.G., Li, J.Y.: A new derivative-free SCG-type projection method for nonlinear monotone equations with convex constraints. J. Appl. Math. Comput. 56, 195–216 (2018)

    Article  MathSciNet  Google Scholar 

  13. Zhou, W.J., Wang, F.: A PRP-based residual method for large-scale monotone nonlinear equations. Appl. Math. Comput. 261, 1–7 (2015)

    MathSciNet  Google Scholar 

  14. Zheng, L., Yang, L., Liang, Y.: A conjugate gradient projection method for solving equations with convex constraints. J. Comput. Appl. Math. 375, 112781 (2020)

    Article  MathSciNet  Google Scholar 

  15. Sun, M., Liu, J.: New hybrid conjugate gradient projection method for the convex constrained equations. Calcolo 53(3), 399–411 (2016)

    Article  MathSciNet  Google Scholar 

  16. 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(1), 310–319 (2013)

    Article  MathSciNet  Google Scholar 

  17. Ibrahim, A.H., Kumam, P., Abubakar, A.B., Adamu, A.: Accelerated derivative-free method for nonlinear monotone equations with an application. Numer. Linear. Algebr. 29(3), e2424 (2022)

    Article  MathSciNet  Google Scholar 

  18. Abubakar, A.B., Kumam, P., Ibrahim, A.H., Chaipunya, P., Rano, S.A.: New hybrid three-term spectral-conjugate gradient method for finding solutions of nonlinear monotone operator equations with applications. Math. Comput. Simulat. 201, 670–683 (2022)

    Article  MathSciNet  Google Scholar 

  19. Abubakar, A.B., Kumam, P., Mohammad, A.H., Ibrahim, A.H., Kiri, A.I.: A hybrid approach for finding approximate solutions to constrained nonlinear monotone operator equations with applications. Appl. Numer. Math. 177, 79–92 (2022)

    Article  MathSciNet  Google Scholar 

  20. Wu, X.Y., Shao, H., Liu, P.J., Zhuo, Y.: An inertial spectral CG projeciton method based on the memoryless BFGS update. J. Optimiz. Theory. App. 198, 1130–1155 (2023)

    Article  Google Scholar 

  21. Abdullahi, M., Abubakar, A.B., Feng, Y.M., Liu, J.K.: Comment on: a derivative-free iterative method for nonlinear monotone equations with convex constraints. Numer. Algorithms (2023). https://doi.org/10.1007/s11075-023-01546-5

    Article  MathSciNet  Google Scholar 

  22. Liu, J.K., Sun, Y., Zhao, Y.X.: A derivative-free projection algorithm for solving pseudo monotone equations with convex constraints (in Chinese). Math. Numer. Sin. 43(03), 388–400 (2021)

    MathSciNet  Google Scholar 

  23. Zhang, N., Liu, J.K.: A self-adaptive projection method for nonlinear monotone equations with convex constraints. J. Ind. Manag. Optim. 19(11), 8152–8163 (2023)

    Article  MathSciNet  Google Scholar 

  24. Liu, P.J., Shao, H., Wang, Y., Wu, X.Y.: A three-term CGPM-based algorithm without Lipschitz continuity for constrained nonlinear monotone equations with applications. Appl. Numer. Math. 175, 98–107 (2022)

    Article  MathSciNet  Google Scholar 

  25. Ma, G.D., Lin, H., Jin, W.H., Han, D.L.: Two modified conjugate gradient methods for unconstrained optimization with applications in image restoration problems. J. Appl. Math. Comput. 68, 4733–4758 (2022)

    Article  MathSciNet  Google Scholar 

  26. Abubakar, A.B., Kumam, P., Malik, M., Ibrahim, A.H.: A hybrid conjugate gradient based approach for solving unconstrained optimization and motion control problems. Math. Comput. Simulat. 201, 640–657 (2022)

    Article  MathSciNet  Google Scholar 

  27. Liu, Y.F., Zhu, Z.B., Zhang, B.X.: Two sufficient descent three-term conjugate gradient methods for unconstrained optimization problems with applications in compressive sensing. J. Appl. Math. Comput. 68, 1787–1816 (2022)

    Article  MathSciNet  Google Scholar 

  28. Narushima, Y., Yabe, H., Ford, J.A.: three-term conjugate gradient method with sufficient descent property for unconstrained optimization. SIAM J. Optimiz. 21(1), 212–230 (2011)

    Article  MathSciNet  Google Scholar 

  29. Jiang, X.Z., Yang, H.H., Yin, J.H., Liao, W.: A three-term conjugate gradient algorithm with restart procedure to solve image restoration problems. J. Comput. Appl. Math. 424, 115020 (2023)

    Article  MathSciNet  Google Scholar 

  30. Zhang, L., Zhou, W.J.: Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196(2), 478–484 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  31. Zhou, W.J., Li, D.H.: A globally convergent BFGS method for nonlinear monotone equations without any merit functions. Math. Comput. 77(264), 2231–2240 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  32. Amini, K., Kamandi, A.: A new line search strategy for finding separating hyperplane in projection-based methods. Numer. Algorithms 70(3), 559–570 (2015)

    Article  MathSciNet  Google Scholar 

  33. Yin, J.H., Jian, J.B., Jiang, X.Z., Liu, M.X., Wang, L.Z.: A hybrid three-term conjugate gradient projection method for constrained nonlinear monotone equations with applications. Numer. Algorithms 88, 389–418 (2021)

    Article  MathSciNet  Google Scholar 

  34. Zarantonello, E.H.: Projections on Convex Sets in Hilbert Space and Spectral Theory. Academic Press, New York (1971)

    Google Scholar 

  35. Ibrahim, A.H., Kumam, P., Sun, M., Chaipunya, P., Abubakar, A.B.: Projection method with inertial step for nonlinear equations: application to Signal Recovery. J. Ind. Manag. Optim. 19(1), 30–55 (2023)

    Article  MathSciNet  Google Scholar 

  36. Ma, G.D., Jin, J.C., Jian, J.B., Yin, J.H., Han, D.L.: A modified inertial three-term conjugate gradient projection method for constrained nonlinear equations with applications in compressed sensing. Numer. Algorithms 92(3), 1621–1653 (2023)

    Article  MathSciNet  Google Scholar 

  37. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

    Article  MathSciNet  Google Scholar 

  38. Banham, M.R., Katsaggelos, A.K.: Digital image restoration. IEEE Signal. Proc. Mag. 14(2), 24–41 (1997)

    Article  Google Scholar 

  39. Chan, C.L., Katsaggelos, A.K., Sahakian, A.V.: Image sequence filtering in quantum-limited noise with applications to low-dose fluoroscopy. IEEE T. Med. Imaging 12(3), 610–621 (1993)

    Article  CAS  Google Scholar 

  40. Figueiredo, M.A., Nowak, R.D., Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J-STSP. 1(4), 586–597 (2007)

    Google Scholar 

  41. Xiao, Y.H., Wang, Q.Y., Hu, Q.J.: Non-smooth equations based method for \(\ell _{1}\)-norm problems with applications to compressed sensing. Nonlinear. Anal-Theor. 74(11), 3570–3577 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the National Natural Science Foundation of China (12261008), the Natural Science Foundation of Guangxi Province (2023GXNSFAA026158), the Xiangsihu Young Scholars Innovative Research Team of Guangxi Minzu University(2022GXUNXSHQN04) and the Guangxi Scholarship Fund of Guangxi Education Department (GED)

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi Minzu University, Nanning, 530006, Guangxi, China

    Guodong Ma, Liqi Liu & Jinbao Jian

  2. School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong, 030619, Shanxi, China

    Xihong Yan

Authors
  1. Guodong Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liqi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinbao Jian
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xihong Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinbao Jian.

Ethics declarations

Conflict of interest

The authors declare that they have 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

Ma, G., Liu, L., Jian, J. et al. A new hybrid CGPM-based algorithm for constrained nonlinear monotone equations with applications. J. Appl. Math. Comput. 70, 103–147 (2024). https://doi.org/10.1007/s12190-023-01960-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-023-01960-x

Keywords

Mathematics Subject Classification

Search

Navigation