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On the Quality of First-Order Approximation of Functions with Hölder Continuous Gradient

Author

Listed:
  • Guillaume O. Berger

    (UCLouvain)

  • P.-A. Absil

    (UCLouvain)

  • Raphaël M. Jungers

    (UCLouvain)

  • Yurii Nesterov

    (UCLouvain)

Abstract

We show that Hölder continuity of the gradient is not only a sufficient condition, but also a necessary condition for the existence of a global upper bound on the error of the first-order Taylor approximation. We also relate this global upper bound to the Hölder constant of the gradient. This relation is expressed as an interval, depending on the Hölder constant, in which the error of the first-order Taylor approximation is guaranteed to be. We show that, for the Lipschitz continuous case, the interval cannot be reduced. An application to the norms of quadratic forms is proposed, which allows us to derive a novel characterization of Euclidean norms.

Suggested Citation

  • Guillaume O. Berger & P.-A. Absil & Raphaël M. Jungers & Yurii Nesterov, 2020. "On the Quality of First-Order Approximation of Functions with Hölder Continuous Gradient," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 17-33, April.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:1:d:10.1007_s10957-020-01632-x
    DOI: 10.1007/s10957-020-01632-x
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, 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).
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