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

Guillaume O. Berger (), P.-A. Absil (), Raphaël M. Jungers () and Yurii Nesterov ()
Additional contact information
Guillaume O. Berger: UCLouvain
P.-A. Absil: UCLouvain
Raphaël M. Jungers: UCLouvain
Yurii Nesterov: UCLouvain

Journal of Optimization Theory and Applications, 2020, vol. 185, issue 1, No 2, 17-33

Abstract: 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; 68Q25; 90C30; 90C48 (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10957-020-01632-x Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:185:y:2020:i:1:d:10.1007_s10957-020-01632-x

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-020-01632-x

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().