Smoothness Parameter of Power of Euclidean Norm

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 cas...

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Detalles Bibliográficos
Autores principales: Rodomanov, Anton, Nesterov, Yurii
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2020
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7210250/
https://www.ncbi.nlm.nih.gov/pubmed/32421041
http://dx.doi.org/10.1007/s10957-020-01653-6
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author Rodomanov, Anton
Nesterov, Yurii
author_facet Rodomanov, Anton
Nesterov, Yurii
author_sort Rodomanov, Anton
collection PubMed
description 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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spelling pubmed-72102502020-05-13 Smoothness Parameter of Power of Euclidean Norm Rodomanov, Anton Nesterov, Yurii J Optim Theory Appl Article 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. Springer US 2020-03-27 2020 /pmc/articles/PMC7210250/ /pubmed/32421041 http://dx.doi.org/10.1007/s10957-020-01653-6 Text en © The Author(s) 2020 Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
spellingShingle Article
Rodomanov, Anton
Nesterov, Yurii
Smoothness Parameter of Power of Euclidean Norm
title Smoothness Parameter of Power of Euclidean Norm
title_full Smoothness Parameter of Power of Euclidean Norm
title_fullStr Smoothness Parameter of Power of Euclidean Norm
title_full_unstemmed Smoothness Parameter of Power of Euclidean Norm
title_short Smoothness Parameter of Power of Euclidean Norm
title_sort smoothness parameter of power of euclidean norm
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7210250/
https://www.ncbi.nlm.nih.gov/pubmed/32421041
http://dx.doi.org/10.1007/s10957-020-01653-6
work_keys_str_mv AT rodomanovanton smoothnessparameterofpowerofeuclideannorm
AT nesterovyurii smoothnessparameterofpowerofeuclideannorm

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