Search Advanced

AIMS Mathematics

2022, Volume 7, Issue 1: 1241-1256. doi: 10.3934/math.2022073
Previous Article Next Article
Research article

A variant of the Levenberg-Marquardt method with adaptive parameters for systems of nonlinear equations

  • 1.
    School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, Anhui 233030, China
  • 2.
    School of Sciences, Changzhou Institute of Technology, Changzhou, Jiangsu 213032, China
  • 3.
    School of Computer Science and Information Engineering, Changzhou Institute of Technology, Changzhou, Jiangsu 213032, China
  • 4.
    Institute of Quantitative Economics, Anhui University of Finance and Economics, Bengbu, Anhui 233030, China
  • 5.
    School of Computer Science and Technology, Huaibei Normal University, Huaibei, Anhui 235000, China
  • Received: 11 July 2021 Accepted: 11 October 2021 Published: 21 October 2021

  • MSC : 65K05, 90C30

  • The Levenberg-Marquardt method is one of the most important methods for solving systems of nonlinear equations and nonlinear least-squares problems. It enjoys a quadratic convergence rate under the local error bound condition. Recently, to solve nonzero-residue nonlinear least-squares problem, Behling et al. propose a modified Levenberg-Marquardt method with at least superlinearly convergence under a new error bound condtion [3]. To extend their results for systems of nonlinear equations, by choosing the LM parameters adaptively, we propose an efficient variant of the Levenberg-Marquardt method and prove its quadratic convergence under the new error bound condition. We also investigate its global convergence by using the Wolfe line search. The effectiveness of the new method is validated by some numerical experiments.

    Citation: Lin Zheng, Liang Chen, Yanfang Ma. A variant of the Levenberg-Marquardt method with adaptive parameters for systems of nonlinear equations[J]. AIMS Mathematics, 2022, 7(1): 1241-1256. doi: 10.3934/math.2022073

    Related Papers:

  • The Levenberg-Marquardt method is one of the most important methods for solving systems of nonlinear equations and nonlinear least-squares problems. It enjoys a quadratic convergence rate under the local error bound condition. Recently, to solve nonzero-residue nonlinear least-squares problem, Behling et al. propose a modified Levenberg-Marquardt method with at least superlinearly convergence under a new error bound condtion [3]. To extend their results for systems of nonlinear equations, by choosing the LM parameters adaptively, we propose an efficient variant of the Levenberg-Marquardt method and prove its quadratic convergence under the new error bound condition. We also investigate its global convergence by using the Wolfe line search. The effectiveness of the new method is validated by some numerical experiments.


    加载中


    [1] M. Ahookhosh, F. J. A. Artacho, R. M. T. Fleming, P. T. Vuong, Local convergence of the Levenberg–Marquardt method under {H}ölder metric subregularity, Adv. Comput. Math., 45 (2019), 2771–2806. doi: 10.1007/s10444-019-09708-7. doi: 10.1007/s10444-019-09708-7
    [2] K. Amini, F. Rostami, G. Caristi, An efficient Levenberg-Marquardt method with a new LM parameter for systems of nonlinear equations, Optimization, 67 (2018), 637–650. doi: 10.1080/02331934.2018.1435655. doi: 10.1080/02331934.2018.1435655
    [3] R. Behling, D. S. Gonçalves, S. A. Santos, Local convergence analysis of the Levenberg–Marquardt framework for Nonzero-Residue nonlinear least-squares problems under an error bound condition, J. Optim. Theory Appl., 183 (2019), 1099–1122. doi: 10.1007/s10957-019-01586-9. doi: 10.1007/s10957-019-01586-9
    [4] J. E. Dennis, R. B. Schnable, Numerical methods for unconstrained optimization and nonlinear equations, 1983. doi: 10.1137/1.9781611971200.
    [5] J. Y. Fan, J. Y. Pan, A note on the Levenberg-Marquardt parameter, Appl. Math. Comput., 207 (2009), 351–359. doi: 10.1016/j.amc.2008.10.056. doi: 10.1016/j.amc.2008.10.056
    [6] J. Y. Fan, Y. X. Yuan, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing, 74 (2005), 23–39. doi: 10.1007/s00607-004-0083-1. doi: 10.1007/s00607-004-0083-1
    [7] F. Andreas, Local behavior of an iterative framework for generalized equations with nonisolated solutions, Math. Program., 94 (2002), 91–124. doi: 10.1007/s10107-002-0364-4. doi: 10.1007/s10107-002-0364-4
    [8] L. Guo, G. H. Lin, J. J. Ye, Solving mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 166 (2015), 234–256. doi: 10.1007/s10957-014-0699-z. doi: 10.1007/s10957-014-0699-z
    [9] M. Heydari, T. D. Niri, M. M. Hosseini, A new modified trust region algorithm for solving unconstrained optimization problems, J. Math. Ext., 12 (2018), 115–135.
    [10] E. W. Karas, S. A. Santos, B. F. Svaiter, Algebraic rules for computing the regularization parameter of the Levenberg–Marquardt method, Comput. Optim. Appl., 65 (2016), 723–751. doi: 10.1007/s10589-016-9845-x. doi: 10.1007/s10589-016-9845-x
    [11] C. F. Ma, L. H. Jiang, Some research on Levenberg-Marquardt method for the nonlinear equations, Appl. Math. Comput., 184 (2007), 1032–1040. doi: 10.1016/j.amc.2006.07.004. doi: 10.1016/j.amc.2006.07.004
    [12] J. J. Moré, B. S. Garbow, K. E. Hillstrom, Testing unconstrained optimization software, ACM T. Math. Software, 7 (1981), 17–41. doi: 10.1145/355934.355936. doi: 10.1145/355934.355936
    [13] T. D. Niri, M. Heydari, M. M. Hosseini, Correction of trust region method with a new modified Newton method, Int. J. Comput. Math., 97 (2020), 1118–1132. doi: 10.1080/00207160.2019.1607844. doi: 10.1080/00207160.2019.1607844
    [14] H. V. Ngai, Global error bounds for systems of convex polynomials over polyhedral constraints, SIAM J. Optim., 25 (2015), 521–539. doi: 10.1137/13090599X. doi: 10.1137/13090599X
    [15] J. Nocedal, S. J. Wright, Numerical optimization, New York: Springer, 1999. doi: 10.1007/978-3-540-35447-5.
    [16] R. B. Schnabel, P. D. Frank, Tensor methods for nonlinear equations, SIAM J. Numer. Anal., 21 (1984), 815–843. doi: 10.1137/0721054. doi: 10.1137/0721054
    [17] H. H. Vui, Global Hölderian error bound for nondegenerate polynomials, SIAM J. Optim., 23 (2013), 917–933. doi: 10.1137/110859889. doi: 10.1137/110859889
    [18] H. Y. Wang, J. Y. Fan, Convergence rate of the Levenberg-Marquardt method under Hölderian local error bound, Optim. Method. Softw, 35 (2020), 767–786. doi: 10.1080/10556788.2019.1694927. doi: 10.1080/10556788.2019.1694927
    [19] N. Yamashita, M. Fukushima, On the rate of convergence of the Levenberg-Marquardt method, In: Topics in numerical analysis, Vienna: Springer, 15 (2001), 239–249. doi: 10.1007/978-3-7091-6217-0_18.
    [20] Y. X. Yuan, Problems on convergence of unconstrained optimization algorithms, In: Numerical linear algebra and optimization, Beijing: Science Press, 1999, 95–107.
    [21] X. D. Zhu, G. H. Lin, Improved convergence results for a modified Levenberg-Marquardt method for nonlinear equations and applications in MPCC, Optim. Method. Software, 31 (2016), 791–804. doi: 10.1080/10556788.2016.1171863. doi: 10.1080/10556788.2016.1171863
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(2479) PDF downloads(109) Cited by(5)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(3)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog