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Convergence and Complexity Analysis of a Levenberg–Marquardt Algorithm for Inverse Problems

Author

Listed:
  • El Houcine Bergou

    (Université Paris-Saclay
    KAUST)

  • Youssef Diouane

    (Université de Toulouse)

  • Vyacheslav Kungurtsev

    (Czech Technical University in Prague)

Abstract

The Levenberg–Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified variations of the basic procedure, the algorithm enjoys global convergence, a competitive worst-case iteration complexity rate, and a guaranteed rate of local convergence for both zero and nonzero small residual problems, under suitable assumptions. We introduce a novel Levenberg-Marquardt method that matches, simultaneously, the state of the art in all of these convergence properties with a single seamless algorithm. Numerical experiments confirm the theoretical behavior of our proposed algorithm.

Suggested Citation

  • El Houcine Bergou & Youssef Diouane & Vyacheslav Kungurtsev, 2020. "Convergence and Complexity Analysis of a Levenberg–Marquardt Algorithm for Inverse Problems," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 927-944, June.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:3:d:10.1007_s10957-020-01666-1
    DOI: 10.1007/s10957-020-01666-1
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    References listed on IDEAS

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    1. Kenji Ueda & Nobuo Yamashita, 2012. "Global Complexity Bound Analysis of the Levenberg–Marquardt Method for Nonsmooth Equations and Its Application to the Nonlinear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 450-467, February.
    2. Francisco Facchinei & Andreas Fischer & Markus Herrich, 2013. "A family of Newton methods for nonsmooth constrained systems with nonisolated solutions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 433-443, June.
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    Citations

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    Cited by:

    1. Tao Liu & Di Ouyang & Lianjun Guo & Ruofeng Qiu & Yunfei Qi & Wu Xie & Qiang Ma & Chao Liu, 2023. "Combination of Multigrid with Constraint Data for Inverse Problem of Nonlinear Diffusion Equation," Mathematics, MDPI, vol. 11(13), pages 1-15, June.
    2. Boos, Everton & Gonçalves, Douglas S. & Bazán, Fermín S.V., 2024. "Levenberg-Marquardt method with singular scaling and applications," Applied Mathematics and Computation, Elsevier, vol. 474(C).
    3. Andreas Fischer & Alexey F. Izmailov & Mikhail V. Solodov, 2024. "The Levenberg–Marquardt method: an overview of modern convergence theories and more," Computational Optimization and Applications, Springer, vol. 89(1), pages 33-67, September.
    4. Won-Kwang Park, 2021. "Fast Localization of Small Inhomogeneities from Far-Field Pattern Data in the Limited-Aperture Inverse Scattering Problem," Mathematics, MDPI, vol. 9(17), pages 1-22, August.
    5. E. V. Castelani & R. Lopes & W. V. I. Shirabayashi & F. N. C. Sobral, 2021. "A robust method based on LOVO functions for solving least squares problems," Journal of Global Optimization, Springer, vol. 80(2), pages 387-414, June.
    6. Naoki Marumo & Takayuki Okuno & Akiko Takeda, 2023. "Majorization-minimization-based Levenberg–Marquardt method for constrained nonlinear least squares," Computational Optimization and Applications, Springer, vol. 84(3), pages 833-874, April.
    7. Tao Liu & Zijian Ding & Jiayuan Yu & Wenwen Zhang, 2023. "Parameter Estimation for Nonlinear Diffusion Problems by the Constrained Homotopy Method," Mathematics, MDPI, vol. 11(12), pages 1-12, June.

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