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AIMS Mathematics

2022, Volume 7, Issue 1: 1241-1256. doi: 10.3934/math.2022073
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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.


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