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Newton-type methods near critical solutions of piecewise smooth nonlinear equations

Author

Listed:
  • A. Fischer

    (Technische Universität Dresden)

  • A. F. Izmailov

    (Lomonosov Moscow State University (MSU))

  • M. Jelitte

    (Technische Universität Dresden)

Abstract

It is well-recognized that in the presence of singular (and in particular nonisolated) solutions of unconstrained or constrained smooth nonlinear equations, the existence of critical solutions has a crucial impact on the behavior of various Newton-type methods. On the one hand, it has been demonstrated that such solutions turn out to be attractors for sequences generated by these methods, for wide domains of starting points, and with a linear convergence rate estimate. On the other hand, the pattern of convergence to such solutions is quite special, and allows for a sharp characterization which serves, in particular, as a basis for some known acceleration techniques, and for the proof of an asymptotic acceptance of the unit stepsize. The latter is an essential property for the success of these techniques when combined with a linesearch strategy for globalization of convergence. This paper aims at extensions of these results to piecewise smooth equations, with applications to corresponding reformulations of nonlinear complementarity problems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:80:y:2021:i:2:d:10.1007_s10589-021-00306-2
    DOI: 10.1007/s10589-021-00306-2
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    References listed on IDEAS

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    1. 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.
    2. Andreas Fischer & Alexey F. Izmailov & Mikhail V. Solodov, 2019. "Local Attractors of Newton-Type Methods for Constrained Equations and Complementarity Problems with Nonisolated Solutions," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 140-169, January.
    3. 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.
    4. 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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    Cited by:

    1. A. F. Izmailov & M. V. Solodov, 2024. "On Local Behavior of Newton-Type Methods Near Critical Solutions of Constrained Equations," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1103-1126, November.
    2. Andreas Fischer & Alexey F. Izmailov & Mario Jelitte, 2023. "Stability of Singular Solutions of Nonlinear Equations with Restricted Smoothness Assumptions," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 1008-1035, March.

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