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Local Attractors of Newton-Type Methods for Constrained Equations and Complementarity Problems with Nonisolated Solutions

Author

Listed:
  • Andreas Fischer

    (Technische Universität Dresden)

  • Alexey F. Izmailov

    (Lomonosov Moscow State University, MSU
    RUDN University)

  • Mikhail V. Solodov

    (IMPA – Instituto de Matemática Pura e Aplicada)

Abstract

For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well defined and necessarily converges to this specific solution (despite degeneracy, and despite that there are other solutions nearby). We note that unlike the common settings of convergence analyses, our assumptions subsume that a local Lipschitzian error bound does not hold for the solution in question. Our results apply to constrained and projected variants of the Gauss–Newton, Levenberg–Marquardt, and LP-Newton methods. Applications to smooth and piecewise smooth reformulations of complementarity problems are also discussed.

Suggested Citation

  • Andreas Fischer & Alexey F. Izmailov & Mikhail V. Solodov, 2019. "Local Attractors of Newton-Type Methods for Constrained Equations and Complementarity Problems with Nonisolated Solutions," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 140-169, January.
    • Handle: RePEc:spr:joptap:v:180:y:2019:i:1:d:10.1007_s10957-018-1297-2
    DOI: 10.1007/s10957-018-1297-2
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    References listed on IDEAS

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    1. Andreas Fischer & Markus Herrich & Alexey Izmailov & Mikhail Solodov, 2016. "Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions," Computational Optimization and Applications, Springer, vol. 63(2), pages 425-459, March.
    2. R. Behling & A. Fischer & M. Herrich & A. Iusem & Y. Ye, 2014. "A Levenberg-Marquardt method with approximate projections," Computational Optimization and Applications, Springer, vol. 59(1), pages 5-26, October.
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    Cited by:

    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.

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