Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Shamanskii-Like Levenberg-Marquardt Method with a New Line Search for Systems of Nonlinear Equations

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

To save the calculations of Jacobian, a multi-step Levenberg-Marquardt method named Shamanskii-like LM method for systems of nonlinear equations was proposed by Fan (2013). Its convergence properties have been proved by using a trust region technique under the local error bound condition. However, the authors wonder whether the similar convergence properties are still true with standard line searches since the direction may not be a descent direction. For this purpose, the authors present a new nonmonotone m-th order Armijo type line search to guarantee the global convergence. Under the same condition as trust region case, the convergence rate also has been shown to be m + 1 by using this line search technique. Numerical experiments show the new algorithm can save much running time for the large scale problems, so it is efficient and promising.

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
$34.99 /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 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. Levenberg K, A method for the solution of certain nonlinear problems in least squares, Quarterly of Applied Mathematics, 1944, 2: 164–168.

    Article  MathSciNet  Google Scholar 

  2. Marquardt D W, An algorithm for least-squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics, 1963, 11: 431–441.

    Article  MathSciNet  Google Scholar 

  3. Yamashita N and Fukushima M, On the Rate of Convergence of the Levenberg-Marquardt Method, Springer Vienna, Vienna, 2001.

    Book  Google Scholar 

  4. Fan J, The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence, Mathematics of Computation, 2012, 81: 447–466.

    Article  MathSciNet  Google Scholar 

  5. Fan J, A shamanskii-like Levenberg-Marquardt method for nonlinear equations, Computational Optimization and Applications, 2013, 56(1): 63–80.

    Article  MathSciNet  Google Scholar 

  6. Yang X, A higher-order Levenberg-Marquardt method for nonlinear equations, Applied Mathematics and Computation, 2013, 219(22): 10682–10694.

    Article  MathSciNet  Google Scholar 

  7. Amini K and Rostami F, A modified two steps levenberg-marquardt method for nonlinear equations, Journal of Computational and Applied Mathematics, 2015, 288: 341–350.

    Article  MathSciNet  Google Scholar 

  8. Chen L, A high-order modified Levenberg-Marquardt method for systems of nonlinear equations with fourth-order convergence, Applied Mathematics and Computation, 2016, 285: 79–93.

    Article  MathSciNet  Google Scholar 

  9. Zhou W, On the convergence of the modified Levenberg-Marquardt method with a nonmonotone second order armijo type line search, Journal of Computational and Applied Mathematics, 2013, 239: 152–161.

    Article  MathSciNet  Google Scholar 

  10. Amini K and Rostami F, Three-steps modified levenberg-marquardt method with a new line search for systems of nonlinear equations, Journal of Computational and Applied Mathematics, 2016, 300: 30–42.

    Article  MathSciNet  Google Scholar 

  11. Chen L, Du C, and Ma Y, The higher-order levenberg-marquardt method with armijo type line search for nonlinear equations, Optimization Methods and Software, 2017, 32(3): 516–533.

    Article  MathSciNet  Google Scholar 

  12. Kelley C T, Iterative Methods for Optimization, Society for Industrial and Applied Mathematics, Philadelphia, 1999.

    Book  Google Scholar 

  13. Dennis J and Moré J J, A characterization of superlinear convergence and its applications to quasi-newton methods, Mathematics of Computation, 1974, 126(28): 549–560.

    Article  MathSciNet  Google Scholar 

  14. Behling R and Iusem A, The effect of calmness on the solution set of systems of nonlinear equations, Mathematical Programming, 2013, 137(1–2): 155–165.

    Article  MathSciNet  Google Scholar 

  15. Stewart G W and Sun J, Matrix Perturbation Theory, Computer Science and Scientific Computing, Academic Press Inc., Boston, MA, 1990.

    Google Scholar 

  16. Moré J J, Barbow B S, and Hillstrom K H, Testing unconstrained optimization software, ACM Trans. Math. Software, 1981, 7: 17–41.

    Article  MathSciNet  Google Scholar 

  17. Schnabel R B and Frank P D, Tensor methods for nonlinear equations, SIAM Journal on Numerical Analysis, 1984, 21(5): 815–843.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Huaibei Normal University, Huaibei, 235000, China

    Liang Chen

  2. School of Computer Science and Technology, Huaibei Normal University, Huaibei, 235000, China

    Yanfang Ma

Authors
  1. Liang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanfang Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liang Chen.

Additional information

This research was supported by the Natural Science Foundation of Anhui Province under Grant No. 1708085MF159, the Natural Science Foundation of the Anhui Higher Education Institutions under Grant Nos. KJ2017A375, KJ2019A0604, and the abroad visiting of excellent young talents in universities of Anhui province under Grant No. GXGWFX2019022.

This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, L., Ma, Y. Shamanskii-Like Levenberg-Marquardt Method with a New Line Search for Systems of Nonlinear Equations. J Syst Sci Complex 33, 1694–1707 (2020). https://doi.org/10.1007/s11424-020-9043-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-020-9043-x

Keywords

Search

Navigation