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. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Authors

Journal

Number of issue

2

Language

English

Pages

355-379

Status

Published

Link

Volume

167

Year

2018

Organizations

^{1}VMK Faculty, OR Department, Lomonosov Moscow State University (MSU), Uchebniy Korpus 2, Leninskiye Gory, Moscow, 119991, Russian Federation^{2}Peoples’ Friendship University of Russia, Miklukho-Maklaya Str. 6, Moscow, 117198, Russian Federation^{3}Department of Mathematics, Physics and Computer Sciences, Derzhavin Tambov State University, TSU, Internationalnaya 33, Tambov, 392000, Russian Federation^{4}IMPA—Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil

Keywords

2-Regularity; Critical solutions; Levenberg–Marquardt method; Linear-programming-Newton method; Newton method; Stabilized sequential quadratic programming

Date of creation

19.10.2018

Date of change

19.10.2018

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