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Damped Newton’s method on Riemannian manifolds

Author

Listed:
  • Marcio Antônio de A. Bortoloti

    (DCET, Universidade Estadual do Sudoeste da Bahia)

  • Teles A. Fernandes

    (DCET, Universidade Estadual do Sudoeste da Bahia)

  • Orizon P. Ferreira

    (IME, Universidade Federal de Goiás)

  • Jinyun Yuan

    (Dongguan University of Technology)

Abstract

A damped Newton’s method to find a singularity of a vector field in Riemannian setting is presented with global convergence study. It is ensured that the sequence generated by the proposed method reduces to a sequence generated by the Riemannian version of the classical Newton’s method after a finite number of iterations, consequently its convergence rate is superlinear/quadratic. Even at an early stage of development, we can observe from numerical experiments that DNM presented promising results when compared with the well known BFGS and Trust Regions methods. Moreover, damped Newton’s method present better performance than the Newton’s method in number of iteration and computational time.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:77:y:2020:i:3:d:10.1007_s10898-020-00885-0
    DOI: 10.1007/s10898-020-00885-0
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    References listed on IDEAS

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    1. 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.
    2. 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.
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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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