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On Local Behavior of Newton-Type Methods Near Critical Solutions of Constrained Equations

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Abstract

For constrained equations with nonisolated solutions and a certain family of Newton-type methods, it was previously shown that if the equation mapping is 2-regular at a given solution with respect to a direction which is interior feasible and which is in the null space of the Jacobian, then there is an associated large (not asymptotically thin) domain of starting points from which the iterates are well defined and converge to the specific solution in question. Under these assumptions, the constrained local Lipschitzian error bound does not hold, unlike the common settings of convergence and rate of convergence analyses. In this work, we complement those previous results by considering the case when the equation mapping is 2-regular with respect to a direction in the null space of the Jacobian which is in the tangent cone to the set, but need not be interior feasible. Under some further conditions, we still show linear convergence of order 1/2 from a large domain around the solution (despite degeneracy, and despite that there may exist other solutions nearby). Our results apply to constrained variants of the Gauss–Newton and Levenberg–Marquardt methods, and to the LP-Newton method. An illustration for a smooth constrained reformulation of the nonlinear complementarity problem is also provided.

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Acknowledgements

This research was supported by the Russian Science Foundation Grant 23-11-20020 (https://rscf.ru/en/project/23-11-20020/), by CNPq Grant number 303913/2019-3, by FAPERJ Grant E-26/200.347/2023, and by PRONEX–Optimization.

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Authors and Affiliations

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

    A. F. Izmailov

  2. Derzhavin Tambov State University (TSU), Internationalnaya 33, Tambov, Russia, 392000

    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. F. Izmailov
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Correspondence to M. V. Solodov.

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Communicated by Giovanni Colombo.

Dedicated to Professor Boris Mordukhovich on the occasion of his 75th birthday.

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Izmailov, A.F., Solodov, M.V. On Local Behavior of Newton-Type Methods Near Critical Solutions of Constrained Equations. J Optim Theory Appl (2024). https://doi.org/10.1007/s10957-023-02367-1

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