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On the use of iterative methods in cubic regularization for unconstrained optimization

Author

Listed:
  • Tommaso Bianconcini
  • Giampaolo Liuzzi
  • Benedetta Morini
  • Marco Sciandrone

Abstract

In this paper we consider the problem of minimizing a smooth function by using the adaptive cubic regularized (ARC) framework. We focus on the computation of the trial step as a suitable approximate minimizer of the cubic model and discuss the use of matrix-free iterative methods. Our approach is alternative to the implementation proposed in the original version of ARC, involving a linear algebra phase, but preserves the same worst-case complexity count. Further we introduce a new stopping criterion in order to properly manage the “over-solving” issue arising whenever the cubic model is not an adequate model of the true objective function. Numerical experiments conducted by using a nonmonotone gradient method as inexact solver are presented. The obtained results clearly show the effectiveness of the new variant of ARC algorithm. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Tommaso Bianconcini & Giampaolo Liuzzi & Benedetta Morini & Marco Sciandrone, 2015. "On the use of iterative methods in cubic regularization for unconstrained optimization," Computational Optimization and Applications, Springer, vol. 60(1), pages 35-57, January.
    • Handle: RePEc:spr:coopap:v:60:y:2015:i:1:p:35-57
    DOI: 10.1007/s10589-014-9672-x
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2007. "Gauss-Newton scheme with worst case guarantees for global performance," LIDAM Reprints CORE 1952, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. 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.
    3. Hande Benson & David Shanno, 2014. "Interior-point methods for nonconvex nonlinear programming: cubic regularization," Computational Optimization and Applications, Springer, vol. 58(2), pages 323-346, June.
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    Citations

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    Cited by:

    1. J. M. Martínez & M. Raydan, 2017. "Cubic-regularization counterpart of a variable-norm trust-region method for unconstrained minimization," Journal of Global Optimization, Springer, vol. 68(2), pages 367-385, June.
    2. J. Martínez & M. Raydan, 2015. "Separable cubic modeling and a trust-region strategy for unconstrained minimization with impact in global optimization," Journal of Global Optimization, Springer, vol. 63(2), pages 319-342, October.
    3. Rujun Jiang & Man-Chung Yue & Zhishuo Zhou, 2021. "An accelerated first-order method with complexity analysis for solving cubic regularization subproblems," Computational Optimization and Applications, Springer, vol. 79(2), pages 471-506, June.

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