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Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method

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  • Nikita Doikov

    (ICTEAM (Catholic University of Louvain))

  • Yurii Nesterov

    (CORE (Catholic University of Louvain))

Abstract

In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically achieves the best possible global complexity bound among different problem classes of uniformly convex objective functions with Hölder continuous Hessian of the smooth part of the objective. As a byproduct of our developments, we justify an intuitively plausible result that the global iteration complexity of the Newton method is always better than that of the gradient method on the class of strongly convex functions with uniformly bounded second derivative.

Suggested Citation

  • Nikita Doikov & Yurii Nesterov, 2021. "Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 317-339, April.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:1:d:10.1007_s10957-021-01838-7
    DOI: 10.1007/s10957-021-01838-7
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    References listed on IDEAS

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    1. 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).
    2. 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).
    3. 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).
    4. Geovani N. Grapiglia & Yurii Nesterov, 2019. "Accelerated regularized Newton methods for minimizing composite convex functions," LIDAM Reprints CORE 3058, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Doikov, Nikita & Nesterov, Yurii, 2021. "Optimization Methods for Fully Composite Problems," LIDAM Discussion Papers CORE 2021001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Aghaseyedabdollah, Mohammadhossein & Abedi, Mostafa & Pourgholi, Mahdi, 2024. "Interval type-2 fuzzy sliding mode control for a cable-driven parallel robot with elastic cables using metaheuristic optimization methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 435-461.

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