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Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition

Author

Listed:
  • Roger Behling

    (Fundação Getúlio Vargas)

  • Douglas S. Gonçalves

    (Federal University of Santa Catarina)

  • Sandra A. Santos

    (University of Campinas)

Abstract

The Levenberg–Marquardt method is widely used for solving nonlinear systems of equations, as well as nonlinear least-squares problems. In this paper, we consider local convergence properties of the method, when applied to nonzero-residue nonlinear least-squares problems under an error bound condition, which is weaker than requiring full rank of the Jacobian in a neighborhood of a stationary point. Differently from the zero-residue case, the choice of the Levenberg–Marquardt parameter is shown to be dictated by (i) the behavior of the rank of the Jacobian and (ii) a combined measure of nonlinearity and residue size in a neighborhood of the set of (possibly non-isolated) stationary points of the sum of squares function.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:3:d:10.1007_s10957-019-01586-9
    DOI: 10.1007/s10957-019-01586-9
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    References listed on IDEAS

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    1. Elizabeth W. Karas & Sandra A. Santos & Benar F. Svaiter, 2016. "Algebraic rules for computing the regularization parameter of the Levenberg–Marquardt method," Computational Optimization and Applications, Springer, vol. 65(3), pages 723-751, December.
    2. Stefania Bellavia & Elisa Riccietti, 2018. "On an Elliptical Trust-Region Procedure for Ill-Posed Nonlinear Least-Squares Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 824-859, September.
    3. Jean-Baptiste Hiriart-Urruty & Hai Le, 2013. "A variational approach of the rank function," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 207-240, July.
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    Citations

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    Cited by:

    1. 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.
    2. Jianghua Yin & Jinbao Jian & Guodong Ma, 2024. "A modified inexact Levenberg–Marquardt method with the descent property for solving nonlinear equations," Computational Optimization and Applications, Springer, vol. 87(1), pages 289-322, January.
    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. 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).
    5. Reza Arefidamghani & Roger Behling & Yunier Bello-Cruz & Alfredo N. Iusem & Luiz-Rafael Santos, 2021. "The circumcentered-reflection method achieves better rates than alternating projections," Computational Optimization and Applications, Springer, vol. 79(2), pages 507-530, June.
    6. Nataša Krejić & Greta Malaspina & Lense Swaenen, 2023. "A split Levenberg-Marquardt method for large-scale sparse problems," Computational Optimization and Applications, Springer, vol. 85(1), pages 147-179, May.

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