Local convergence of quasi-Newton methods under metric regularity

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


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.


Generalized equation Quasi-Newton method Broyden update Strong metric subregularity Metric regularity Strong metric regularity q-Superlinear convergence 



The authors wish to thank the anonymous referees for their valuable comments and suggestions.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • F. J. Aragón Artacho
    • 1
  • A. Belyakov
    • 2
    • 3
  • A. L. Dontchev
    • 4
  • M. López
    • 5
  1. 1.Systems Biochemistry Group, Luxembourg Centre for Systems BiomedicineUniversity of LuxembourgEsch-sur-AlzetteLuxembourg
  2. 2.Institute of Mathematical Methods in EconomicsVienna University of TechnologyViennaAustria
  3. 3.Institute of MechanicsLomonosov Moscow State UniversityMoscowRussia
  4. 4.Mathematical ReviewsAnn ArborUSA
  5. 5.Department of Statistics and Operations ResearchUniversity of AlicanteAlicanteSpain

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