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

Author

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  • A. F. Izmailov

    (Lomonosov Moscow State University
    Derzhavin Tambov State University (TSU))

  • M. V. Solodov

    (IMPA – Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico)

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.

Suggested Citation

  • A. F. Izmailov & M. V. Solodov, 2024. "On Local Behavior of Newton-Type Methods Near Critical Solutions of Constrained Equations," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1103-1126, November.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:2:d:10.1007_s10957-023-02367-1
    DOI: 10.1007/s10957-023-02367-1
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    References listed on IDEAS

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    1. 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.
    2. A. F. Izmailov & M. V. Solodov, 2002. "The Theory of 2-Regularity for Mappings with Lipschitzian Derivatives and its Applications to Optimality Conditions," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 614-635, August.
    3. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
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