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A Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization

Author

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  • E. G. Birgin

    (University of São Paulo)

  • J. M. Martínez

    (State University of Campinas)

Abstract

A Newton-like method for unconstrained minimization is introduced in the present work. While the computer work per iteration of the best-known implementations may need several factorizations or may use rather expensive matrix decompositions, the proposed method uses a single cheap factorization per iteration. Convergence and complexity and results, even in the case in which the subproblems’ Hessians are far from being Hessians of the objective function, are presented. Moreover, when the Hessian is Lipschitz-continuous, the proposed method enjoys $$O(\varepsilon ^{-3/2})$$ O ( ε - 3 / 2 ) evaluation complexity for first-order optimality and $$O(\varepsilon ^{-3})$$ O ( ε - 3 ) for second-order optimality as other recently introduced Newton method for unconstrained optimization based on cubic regularization or special trust-region procedures. Fairly successful and fully reproducible numerical experiments are presented and the developed corresponding software is freely available.

Suggested Citation

  • E. G. Birgin & J. M. Martínez, 2019. "A Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization," Computational Optimization and Applications, Springer, vol. 73(3), pages 707-753, July.
    • Handle: RePEc:spr:coopap:v:73:y:2019:i:3:d:10.1007_s10589-019-00089-7
    DOI: 10.1007/s10589-019-00089-7
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    References listed on IDEAS

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    1. E. Bergou & Y. Diouane & S. Gratton, 2017. "On the use of the energy norm in trust-region and adaptive cubic regularization subproblems," Computational Optimization and Applications, Springer, vol. 68(3), pages 533-554, December.
    2. 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.
    3. 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.
    4. Geovani N. GRAPIGLIA & Yurii NESTEROV, 2017. "Regularized Newton methods for minimizing functions with Hölder continuous Hessians," LIDAM Reprints CORE 2846, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Sha Lu & Zengxin Wei & Lue Li, 2012. "A trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization," Computational Optimization and Applications, Springer, vol. 51(2), pages 551-573, March.
    6. Ernesto Birgin & Jan Gentil, 2012. "Evaluating bound-constrained minimization software," Computational Optimization and Applications, Springer, vol. 53(2), pages 347-373, October.
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    Cited by:

    1. J. M. Martínez & L. T. Santos, 2022. "On large-scale unconstrained optimization and arbitrary regularization," Computational Optimization and Applications, Springer, vol. 81(1), pages 1-30, January.
    2. V. S. Amaral & R. Andreani & E. G. Birgin & D. S. Marcondes & J. M. Martínez, 2022. "On complexity and convergence of high-order coordinate descent algorithms for smooth nonconvex box-constrained minimization," Journal of Global Optimization, Springer, vol. 84(3), pages 527-561, November.
    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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