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Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method

Author

Listed:
  • Doikov, Nikita

    (Université catholique de Louvain, ICTEAM)

  • Nesterov, Yurii

    (Université catholique de Louvain, LIDAM/CORE, Belgium)

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

  • Doikov, Nikita & Nesterov, Yurii, 2023. "Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method," LIDAM Reprints CORE 3235, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:3235
    DOI: https://doi.org/10.1007/s10957-021-01838-7
    Note: In: Journal of Optimization Theory and Applications, 2021, vol. 189(1), p. 317-339
    as

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