Kantorovich’s theorem on Newton’s method under majorant condition in Riemannian manifolds

T. Bittencourt () and O. P. Ferreira ()
Additional contact information
T. Bittencourt: IME/UFG
O. P. Ferreira: IME/UFG

Journal of Global Optimization, 2017, vol. 68, issue 2, No 8, 387-411

Abstract: Abstract Extension of concepts and techniques of linear spaces for the Riemannian setting has been frequently attempted. One reason for the extension of such techniques is the possibility to transform some Euclidean non-convex or quasi-convex problems into Riemannian convex problems. In this paper, a version of Kantorovich’s theorem on Newton’s method for finding a singularity of differentiable vector fields defined on a complete Riemannian manifold is presented. In the presented analysis, the classical Lipschitz condition is relaxed using a general majorant function, which enables us to not only establish the existence and uniqueness of the solution but also unify earlier results related to Newton’s method. Moreover, a ball is prescribed around the points satisfying Kantorovich’s assumptions and convergence of the method is ensured for any starting point within this ball. In addition, some bounds for the Q-quadratic convergence of the method, which depends on the majorant function, are obtained.

Keywords: Newton’s method; Robust Kantorovich’s theorem; Majorant function; Vector field; Riemannian manifold (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10898-016-0472-y 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:jglopt:v:68:y:2017:i:2:d:10.1007_s10898-016-0472-y

Ordering information: This journal article can be ordered from
http://www.springer. ... search/journal/10898

DOI: 10.1007/s10898-016-0472-y

Access Statistics for this article

Journal of Global Optimization is currently edited by Sergiy Butenko

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