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

Guillaume Berger, Pierre-Antoine Absil, Raphaël M. Jungers and Yurii Nesterov
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Yurii Nesterov: Université catholique de Louvain, LIDAM/CORE, Belgium

No 3176, LIDAM Reprints CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

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.

Keywords: Hölder continuous gradient; First-order Taylor approximation; Lipschitz continuous gradient; Lipschitz constant; Euclidean norms (search for similar items in EconPapers)
Pages: 17
Date: 2021-01-01
Note: In: Journal of Optimization Theory and Applications, 2020, vol. 185, p. 17-33
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Persistent link: https://EconPapers.repec.org/RePEc:cor:louvrp:3176

DOI: 10.1007/s10957-020-01632-x

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