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Local convergence of quasi-Newton methods under metric regularity

Author

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  • F. Aragón Artacho
  • A. Belyakov
  • A. Dontchev
  • M. López

Abstract

We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • F. Aragón Artacho & A. Belyakov & A. Dontchev & M. López, 2014. "Local convergence of quasi-Newton methods under metric regularity," Computational Optimization and Applications, Springer, vol. 58(1), pages 225-247, May.
    • Handle: RePEc:spr:coopap:v:58:y:2014:i:1:p:225-247
    DOI: 10.1007/s10589-013-9615-y
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    References listed on IDEAS

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    1. Boubakeur Benahmed & Hocine Mokhtar-Kharroubi & Bruno Malafosse & Adnan Yassine, 2011. "Quasi-Newton methods in infinite-dimensional spaces and application to matrix equations," Journal of Global Optimization, Springer, vol. 49(3), pages 365-379, March.
    2. Wenyu Sun & Ya-Xiang Yuan, 2006. "Optimization Theory and Methods," Springer Optimization and Its Applications, Springer, number 978-0-387-24976-6, December.
    3. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
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    Cited by:

    1. Jiaxi Wang & Wei Ouyang, 2022. "Newton’s Method for Solving Generalized Equations Without Lipschitz Condition," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 510-532, February.
    2. Wei Ouyang & Kui Mei, 2023. "A General Iterative Procedure for Solving Nonsmooth Constrained Generalized Equations," Mathematics, MDPI, vol. 11(22), pages 1-17, November.
    3. Michaël Gaydu & Gilson N. Silva, 2020. "A General Iterative Procedure to Solve Generalized Equations with Differentiable Multifunction," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 207-222, April.
    4. Fabiana R. Oliveira & Orizon P. Ferreira & Gilson N. Silva, 2019. "Newton’s method with feasible inexact projections for solving constrained generalized equations," Computational Optimization and Applications, Springer, vol. 72(1), pages 159-177, January.

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