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A family of Newton methods for nonsmooth constrained systems with nonisolated solutions

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  • Francisco Facchinei
  • Andreas Fischer
  • Markus Herrich

Abstract

We propose a new family of Newton-type methods for the solution of constrained systems of equations. Under suitable conditions, that do not include differentiability or local uniqueness of solutions, local, quadratic convergence to a solution of the system of equations can be established. We show that as particular instances of the method we obtain inexact versions of both a recently introduced LP-based Newton method and of a Levenberg-Marquardt algorithm for the solution of systems with nonisolated solutions, and improve on corresponding existing results. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:77:y:2013:i:3:p:433-443
    DOI: 10.1007/s00186-012-0419-0
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    Citations

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    Cited by:

    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. 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.
    3. Francisco Facchinei & Christian Kanzow & Sebastian Karl & Simone Sagratella, 2015. "The semismooth Newton method for the solution of quasi-variational inequalities," Computational Optimization and Applications, Springer, vol. 62(1), pages 85-109, September.
    4. A. Fischer & M. Herrich & A. F. Izmailov & W. Scheck & M. V. Solodov, 2018. "A globally convergent LP-Newton method for piecewise smooth constrained equations: escaping nonstationary accumulation points," Computational Optimization and Applications, Springer, vol. 69(2), pages 325-349, March.
    5. Andreas Fischer & Markus Herrich & Alexey Izmailov & Mikhail Solodov, 2016. "Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions," Computational Optimization and Applications, Springer, vol. 63(2), pages 425-459, March.
    6. A. F. Izmailov & M. V. Solodov & E. I. Uskov, 2019. "A globally convergent Levenberg–Marquardt method for equality-constrained optimization," Computational Optimization and Applications, Springer, vol. 72(1), pages 215-239, January.
    7. A. Fischer & A. F. Izmailov & M. Jelitte, 2021. "Newton-type methods near critical solutions of piecewise smooth nonlinear equations," Computational Optimization and Applications, Springer, vol. 80(2), pages 587-615, November.
    8. 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.
    9. 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.
    10. A. Fischer & A. F. Izmailov & M. V. Solodov, 2021. "Accelerating convergence of the globalized Newton method to critical solutions of nonlinear equations," Computational Optimization and Applications, Springer, vol. 78(1), pages 273-286, January.

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