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On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds

Author

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  • Teles A. Fernandes

    (Universidade Estadual do Sudoeste da Bahia)

  • Orizon P. Ferreira

    (Universidade Federal de Goiás)

  • Jinyun Yuan

    (Universidade Federal do Paraná)

Abstract

In this paper, we study Newton’s method for finding a singularity of a differentiable vector field defined on a Riemannian manifold. Under the assumption of invertibility of the covariant derivative of the vector field at its singularity, we show that Newton’s method is well defined in a suitable neighborhood of this singularity. Moreover, we show that the sequence generated by Newton’s method converges to the solution with superlinear rate.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:3:d:10.1007_s10957-017-1107-2
    DOI: 10.1007/s10957-017-1107-2
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    References listed on IDEAS

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    1. 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.
    2. David G. Luenberger, 1972. "The Gradient Projection Method Along Geodesics," Management Science, INFORMS, vol. 18(11), pages 620-631, July.
    3. P.-A. Absil & Luca Amodei & Gilles Meyer, 2014. "Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries," Computational Statistics, Springer, vol. 29(3), pages 569-590, June.
    4. NESTEROV , Yu. & TODD, Mike, 2002. "On the Riemannian geometry defined by self-concordant barriers and interior-point methods," LIDAM Reprints CORE 1595, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. 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.
    2. Marcio Antônio de A. Bortoloti & Teles A. Fernandes & Orizon P. Ferreira & Jinyun Yuan, 2020. "Damped Newton’s method on Riemannian manifolds," Journal of Global Optimization, Springer, vol. 77(3), pages 643-660, July.
    3. 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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