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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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  • Fabiana R. de Oliveira

    (IME, Universidade Federal de Goiás)

  • Orizon P. Ferreira

    (IME, Universidade Federal de Goiás)

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.

Suggested Citation

  • Fabiana R. de Oliveira & Orizon P. Ferreira, 2020. "Newton Method for Finding a Singularity of a Special Class of Locally Lipschitz Continuous Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 522-539, May.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:2:d:10.1007_s10957-020-01656-3
    DOI: 10.1007/s10957-020-01656-3
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    References listed on IDEAS

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    1. O. Ferreira & A. Iusem & S. Németh, 2013. "Projections onto convex sets on the sphere," Journal of Global Optimization, Springer, vol. 57(3), pages 663-676, November.
    2. Teles A. Fernandes & Orizon P. Ferreira & Jinyun Yuan, 2017. "On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 828-843, June.
    3. S. Hosseini & M. R. Pouryayevali, 2013. "Nonsmooth Optimization Techniques on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 328-342, August.
    4. J. H. Wang, 2011. "Convergence of Newton’s Method for Sections on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 125-145, January.
    5. O. Ferreira & A. Iusem & S. Németh, 2014. "Concepts and techniques of optimization on the sphere," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 1148-1170, October.
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    Cited by:

    1. Fabiana R. Oliveira & Fabrícia R. Oliveira, 2021. "A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 259-273, July.

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