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Smoothness Parameter of Power of Euclidean Norm

Author

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  • Anton Rodomanov

    (Catholic University of Louvain)

  • Yurii Nesterov

    (Catholic University of Louvain)

Abstract

In this paper, we study derivatives of powers of Euclidean norm. We prove their Hölder continuity and establish explicit expressions for the corresponding constants. We show that these constants are optimal for odd derivatives and at most two times suboptimal for the even ones. In the particular case of integer powers, when the Hölder continuity transforms into the Lipschitz continuity, we improve this result and obtain the optimal constants.

Suggested Citation

  • Anton Rodomanov & Yurii Nesterov, 2020. "Smoothness Parameter of Power of Euclidean Norm," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 303-326, May.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:2:d:10.1007_s10957-020-01653-6
    DOI: 10.1007/s10957-020-01653-6
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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. NESTEROV, Yurii, 2015. "Universal gradient methods for convex optimization problems," LIDAM Reprints CORE 2701, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. 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).
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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).

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