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Algebraic rules for quadratic regularization of Newton’s method

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  • Elizabeth Karas
  • Sandra Santos
  • Benar Svaiter

Abstract

In this work we propose a class of quasi-Newton methods to minimize a twice differentiable function with Lipschitz continuous Hessian. These methods are based on the quadratic regularization of Newton’s method, with algebraic explicit rules for computing the regularizing parameter. The convergence properties of this class of methods are analysed. We show that if the sequence generated by the algorithm converges then its limit point is stationary. We also establish local quadratic convergence in a neighborhood of a stationary point with positive definite Hessian. Encouraging numerical experiments are presented. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Elizabeth Karas & Sandra Santos & Benar Svaiter, 2015. "Algebraic rules for quadratic regularization of Newton’s method," Computational Optimization and Applications, Springer, vol. 60(2), pages 343-376, March.
    • Handle: RePEc:spr:coopap:v:60:y:2015:i:2:p:343-376
    DOI: 10.1007/s10589-014-9671-y
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    References listed on IDEAS

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    1. Ernesto Birgin & Jan Gentil, 2012. "Evaluating bound-constrained minimization software," Computational Optimization and Applications, Springer, vol. 53(2), pages 347-373, October.
    2. Marc Fuentes & Jérôme Malick & Claude Lemaréchal, 2012. "Descentwise inexact proximal algorithms for smooth optimization," Computational Optimization and Applications, Springer, vol. 53(3), pages 755-769, December.
    3. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. N. Gould & M. Porcelli & P. Toint, 2012. "Updating the regularization parameter in the adaptive cubic regularization algorithm," Computational Optimization and Applications, Springer, vol. 53(1), pages 1-22, September.
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