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Local convergence of quasi-Newton methods under metric regularity

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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.

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Acknowledgement

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

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Authors and Affiliations

  1. Systems Biochemistry Group, Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Campus Belval, 4362, Esch-sur-Alzette, Luxembourg

    F. J. Aragón Artacho

  2. Institute of Mathematical Methods in Economics, Vienna University of Technology, 1040, Vienna, Austria

    A. Belyakov

  3. Institute of Mechanics, Lomonosov Moscow State University, 119192, Moscow, Russia

    A. Belyakov

  4. Mathematical Reviews, Ann Arbor, MI, 48107-8604, USA

    A. L. Dontchev

  5. Department of Statistics and Operations Research, University of Alicante, 03080, Alicante, Spain

    M. López

Authors
  1. F. J. Aragón Artacho
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  2. A. Belyakov
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  3. A. L. Dontchev
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  4. M. López
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Corresponding author

Correspondence to A. L. Dontchev.

Additional information

A. Belyakov was supported by the Austrian Science Foundation (FWF) under grant No P 24125-N13.

A.L. Dontchev was supported by NSF Grant DMS 1008341 through the University of Michigan.

M. López was supported by MINECO of Spain, Grant MTM2011-29064-C03-02.

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Aragón Artacho, F.J., Belyakov, A., Dontchev, A.L. et al. Local convergence of quasi-Newton methods under metric regularity. Comput Optim Appl 58, 225–247 (2014). https://doi.org/10.1007/s10589-013-9615-y

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  • DOI: https://doi.org/10.1007/s10589-013-9615-y

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