Smoothness Parameter of Power of Euclidean Norm

Rodomanov, Anton;Nesterov, Yurii
(2020) Journal of Optimization Theory and Applications — Vol. 185, n° 2, p. 303-326 (2020)

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  • Rodomanov, AntonUCLouvain
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  • Nesterov, YuriiUCLouvain
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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.
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Rodomanov, A., & Nesterov, Y. (2020). Smoothness Parameter of Power of Euclidean Norm. Journal of Optimization Theory and Applications, 185(2), 303-326. https://doi.org/10.1007/s10957-020-01653-6 (Original work published 2020)