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

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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.

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Acknowledgments

Funding was provided by Fundação de Apoio a Pesquisa do Estado de Goiás (Grant No. 201210267000909-05/2012), CNPq (Grant No 305158/2014-7), CAPES.

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Authors and Affiliations

  1. IME/UFG, Avenida Esperança, s/n, Campus Samambaia, Goiânia, GO, 74690-900, Brazil

    T. Bittencourt & O. P. Ferreira

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  1. T. Bittencourt
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  2. O. P. Ferreira
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Correspondence to O. P. Ferreira.

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This work was supported by PRONEX-Optimization (FAPERJ/CNPq), CNPq 305158/2014-7, FAPEG, CAPES.

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Bittencourt, T., Ferreira, O.P. Kantorovich’s theorem on Newton’s method under majorant condition in Riemannian manifolds. J Glob Optim 68, 387–411 (2017). https://doi.org/10.1007/s10898-016-0472-y

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