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The Levenberg–Marquardt method: an overview of modern convergence theories and more

Author

Listed:
  • Andreas Fischer

    (Technische Universität Dresden)

  • Alexey F. Izmailov

    (Lomonosov Moscow State University (MSU)
    Derzhavin Tambov State University (TSU))

  • Mikhail V. Solodov

    (IMPA – Instituto de Matemática Pura e Aplicada)

Abstract

The Levenberg–Marquardt method is a fundamental regularization technique for the Newton method applied to nonlinear equations, possibly constrained, and possibly with singular or even nonisolated solutions. We review the literature on the subject, in particular relating to each other various convergence frameworks and results. In this process, the analysis is performed from a unified perspective, and some new results are obtained as well. We discuss smooth and piecewise smooth equations, inexact solution of subproblems, and globalization techniques. Attention is also paid to the LP-Newton method, because of its relations to the Levenberg–Marquardt method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:89:y:2024:i:1:d:10.1007_s10589-024-00589-1
    DOI: 10.1007/s10589-024-00589-1
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    References listed on IDEAS

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    1. 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.
    2. Ariizumi, Shumpei & Yamakawa, Yuya & Yamashita, Nobuo, 2024. "Convergence properties of Levenberg–Marquardt methods with generalized regularization terms," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    3. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    4. Axel Dreves & Francisco Facchinei & Andreas Fischer & Markus Herrich, 2014. "A new error bound result for Generalized Nash Equilibrium Problems and its algorithmic application," Computational Optimization and Applications, Springer, vol. 59(1), pages 63-84, October.
    5. 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.
    6. R. Behling & A. Fischer & M. Herrich & A. Iusem & Y. Ye, 2014. "A Levenberg-Marquardt method with approximate projections," Computational Optimization and Applications, Springer, vol. 59(1), pages 5-26, October.
    7. 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.
    8. Letícia Becher & Damián Fernández & Alberto Ramos, 2023. "A trust-region LP-Newton method for constrained nonsmooth equations under Hölder metric subregularity," Computational Optimization and Applications, Springer, vol. 86(2), pages 711-743, November.
    9. Jifeng Bao & Carisa Kwok Wai Yu & Jinhua Wang & Yaohua Hu & Jen-Chih Yao, 2019. "Modified inexact Levenberg–Marquardt methods for solving nonlinear least squares problems," Computational Optimization and Applications, Springer, vol. 74(2), pages 547-582, November.
    10. Roger Behling & Douglas S. Gonçalves & Sandra A. Santos, 2019. "Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1099-1122, December.
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