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Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions

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  • Andreas Fischer
  • Markus Herrich
  • Alexey Izmailov
  • Mikhail Solodov

Abstract

We consider a class of Newton-type methods that are designed for the difficult case when solutions need not be isolated, and the equation mapping need not be differentiable at the solutions. We show that the only structural assumption needed for rapid local convergence of those algorithms applied to PC $$^1$$ 1 -equations is the piecewise error bound, i.e., a local error bound holding for the branches of the solution set resulting from partitions of the bi-active complementarity indices. The latter error bound is implied by various piecewise constraint qualifications, including relatively weak ones. We apply our results to KKT systems arising from optimization or variational problems, and from generalized Nash equilibrium problems. In the first case, we show convergence if the dual part of the solution is a noncritical Lagrange multiplier, and in the second case convergence follows under a relaxed constant rank condition. In both cases, previously available results are improved. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:63:y:2016:i:2:p:425-459
    DOI: 10.1007/s10589-015-9782-0
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    References listed on IDEAS

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    1. Jerzy Kyparisis, 1990. "Sensitivity Analysis for Nonlinear Programs and Variational Inequalities with Nonunique Multipliers," Mathematics of Operations Research, INFORMS, vol. 15(2), pages 286-298, May.
    2. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    3. Axel Dreves & Francisco Facchinei & Andreas Fischer & Markus Herrich, 2014. "A new error bound result for Generalized Nash Equilibrium Problems and its algorithmic application," Computational Optimization and Applications, Springer, vol. 59(1), pages 63-84, October.
    4. Andreas Fischer, 1999. "Modified Wilson's Method for Nonlinear Programs with Nonunique Multipliers," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 699-727, August.
    5. Francisco Facchinei & Andreas Fischer & Markus Herrich, 2013. "A family of Newton methods for nonsmooth constrained systems with nonisolated solutions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 433-443, June.
    6. Alexey Izmailov & Mikhail Solodov, 2014. "On error bounds and Newton-type methods for generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 59(1), pages 201-218, October.
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    Cited by:

    1. A. Fischer & M. Herrich & A. F. Izmailov & W. Scheck & M. V. Solodov, 2018. "A globally convergent LP-Newton method for piecewise smooth constrained equations: escaping nonstationary accumulation points," Computational Optimization and Applications, Springer, vol. 69(2), pages 325-349, March.
    2. María de los Ángeles Martínez & Damián Fernández, 2019. "On the Local and Superlinear Convergence of a Secant Modified Linear-Programming-Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 993-1010, March.
    3. A. F. Izmailov & M. V. Solodov & E. I. Uskov, 2019. "A globally convergent Levenberg–Marquardt method for equality-constrained optimization," Computational Optimization and Applications, Springer, vol. 72(1), pages 215-239, January.
    4. 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.
    5. 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.
    6. Jun Pei & Zorica Dražić & Milan Dražić & Nenad Mladenović & Panos M. Pardalos, 2019. "Continuous Variable Neighborhood Search (C-VNS) for Solving Systems of Nonlinear Equations," INFORMS Journal on Computing, INFORMS, vol. 31(2), pages 235-250, April.

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