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Behavior of Newton-Type Methods Near Critical Solutions of Nonlinear Equations with Semismooth Derivatives

Author

Listed:
  • Andreas Fischer

    (Technische Universität Dresden)

  • Alexey F. Izmailov

    (Lomonosov Moscow State University
    Derzhavin Tambov State University)

  • Mario Jelitte

    (Technische Universität Dresden)

Abstract

Having in mind singular solutions of smooth reformulations of complementarity problems, arising unavoidably when the solution in question violates strict complementarity, we study the behavior of Newton-type methods near singular solutions of nonlinear equations, assuming that the operator of the equation possesses a strongly semismooth derivative, but is not necessarily twice differentiable. These smoothness restrictions give rise to peculiarities of the analysis and results on local linear convergence and asymptotic acceptance of the full step, the issues addressed in this work. Moreover, we consider not only the basic Newton method, but also some stabilized versions of it intended for tackling singular (including nonisolated) solutions. Applications to nonlinear complementarity problems are also dealt with.

Suggested Citation

  • Andreas Fischer & Alexey F. Izmailov & Mario Jelitte, 2024. "Behavior of Newton-Type Methods Near Critical Solutions of Nonlinear Equations with Semismooth Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 203(3), pages 2179-2205, December.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:3:d:10.1007_s10957-023-02350-w
    DOI: 10.1007/s10957-023-02350-w
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    References listed on IDEAS

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    1. 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.
    2. A. Izmailov & A. Kurennoy, 2014. "On regularity conditions for complementarity problems," Computational Optimization and Applications, Springer, vol. 57(3), pages 667-684, April.
    3. 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.
    4. 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.
    5. 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.
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