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Newton Method for Finding a Singularity of a Special Class of Locally Lipschitz Continuous Vector Fields on Riemannian Manifolds

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Abstract

We extend some results of nonsmooth analysis from the Euclidean context to the Riemannian setting. Particularly, we discuss the concepts and some properties, such as the Clarke generalized covariant derivative, upper semicontinuity, and Rademacher theorem, of locally Lipschitz continuous vector fields on Riemannian settings. In addition, we present a version of the Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields. For mild conditions, we establish the well-definedness and local convergence of the sequence generated using the method in a neighborhood of a singularity.

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Acknowledgements

This work was supported by CAPES, FAPEG, and CNPq Grants 408151/2016-1 and 302473/2017-3.

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Authors and Affiliations

  1. IME, Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    Fabiana R. de Oliveira & Orizon P. Ferreira

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  1. Fabiana R. de Oliveira
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  2. Orizon P. Ferreira
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Correspondence to Orizon P. Ferreira.

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Communicated by Sándor Zoltán Németh.

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de Oliveira, F.R., Ferreira, O.P. Newton Method for Finding a Singularity of a Special Class of Locally Lipschitz Continuous Vector Fields on Riemannian Manifolds. J Optim Theory Appl 185, 522–539 (2020). https://doi.org/10.1007/s10957-020-01656-3

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  • DOI: https://doi.org/10.1007/s10957-020-01656-3

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