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Critical solutions of nonlinear equations: local attraction for Newton-type methods

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Abstract

We show that if the equation mapping is 2-regular at a solution in some nonzero direction in the null space of its Jacobian (in which case this solution is critical; in particular, the local Lipschitzian error bound does not hold), then this direction defines a star-like domain with nonempty interior from which the iterates generated by a certain class of Newton-type methods necessarily converge to the solution in question. This is despite the solution being degenerate, and possibly non-isolated (so that there are other solutions nearby). In this sense, Newtonian iterates are attracted to the specific (critical) solution. Those results are related to the ones due to A. Griewank for the basic Newton method but are also applicable, for example, to some methods developed specially for tackling the case of potentially non-isolated solutions, including the Levenberg–Marquardt and the LP-Newton methods for equations, and the stabilized sequential quadratic programming for optimization.

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Acknowledgements

The authors thank the two anonymous referees for their comments on the original version of this article.

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Authors and Affiliations

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

    A. F. Izmailov

  2. Peoples’ Friendship University of Russia, Miklukho-Maklaya Str. 6, Moscow, 117198, Russia

    A. F. Izmailov

  3. Department of Mathematics, Physics and Computer Sciences, Derzhavin Tambov State University, TSU, Internationalnaya 33, Tambov, 392000, Russia

    A. S. Kurennoy

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

    M. V. Solodov

Authors
  1. A. F. Izmailov
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  2. A. S. Kurennoy
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  3. M. V. Solodov
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Corresponding author

Correspondence to M. V. Solodov.

Additional information

This research is supported in part by the Russian Foundation for Basic Research Grant 17-01-00125, by the Russian Science Foundation Grant 15-11-10021, by the Ministry of Education and Science of the Russian Federation (the Agreement number 02.a03.21.0008), by VolkswagenStiftung Grant 115540, by CNPq Grants PVE 401119/2014-9 and 303724/2015-3, and by FAPERJ.

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Izmailov, A.F., Kurennoy, A.S. & Solodov, M.V. Critical solutions of nonlinear equations: local attraction for Newton-type methods. Math. Program. 167, 355–379 (2018). https://doi.org/10.1007/s10107-017-1128-5

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  • DOI: https://doi.org/10.1007/s10107-017-1128-5

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