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Accelerating convergence of the globalized Newton method to critical solutions of nonlinear equations

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Abstract

In the case of singular (and possibly even nonisolated) solutions of nonlinear equations, while superlinear convergence of the Newton method cannot be guaranteed, local linear convergence from large domains of starting points still holds under certain reasonable assumptions. We consider a linesearch globalization of the Newton method, combined with extrapolation and over-relaxation accelerating techniques, aiming at a speed up of convergence to critical solutions (a certain class of singular solutions). Numerical results indicate that an acceleration is observed indeed.

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Acknowledgements

This research was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 409756759, by the Russian Foundation for Basic Research Grants 19-51-12003 NNIO_a and 20-01-00106, by CNPq Grant 303913/2019-3, by FAPERJ Grant E-26/202.540/2019, by PRONEX–Optimization, and by Volkswagen Foundation. The authors thank Ivan Rodin and Dmitriy Bannikov for their assistance with numerical experiments, and the two anonymous referees for helpful comments on the original version of the paper.

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Authors and Affiliations

  1. Faculty of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany

    A. Fischer

  2. VMK Faculty, OR Department, Lomonosov Moscow State University, MSU, Uchebniy Korpus 2, Leninskiye Gory, Moscow, Russia, 119991

    A. F. Izmailov

  3. IMPA – Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil

    M. V. Solodov

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  1. A. Fischer
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  2. A. F. Izmailov
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  3. M. V. Solodov
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Correspondence to A. F. Izmailov.

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Fischer, A., Izmailov, A.F. & Solodov, M.V. Accelerating convergence of the globalized Newton method to critical solutions of nonlinear equations. Comput Optim Appl 78, 273–286 (2021). https://doi.org/10.1007/s10589-020-00230-x

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  • DOI: https://doi.org/10.1007/s10589-020-00230-x

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