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

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 relati...

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Bibliographic Details
Published in:Journal of optimization theory and applications 2020-04, Vol.185 (1), p.17-33
Main Authors: Berger, Guillaume O., Absil, P.-A., Jungers, Raphaël M., Nesterov, Yurii
Format: Article
Language:English
Subjects:
Applications of Mathematics
Approximation
Calculus of Variations and Optimal Control
Optimization
Continuity (mathematics)
Engineering
Mathematical analysis
Mathematics
Mathematics and Statistics
Norms
Operations Research/Decision Theory
Optimization
Quadratic forms
Theory of Computation
Upper bounds
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Summary: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.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-020-01632-x