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A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds

Author

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  • Fabiana R. Oliveira

    (Universidade Federal de Goiás)

  • Fabrícia R. Oliveira

    (Universidade Federal de Goiás)

Abstract

This paper proposes and analyzes a globalized version of the Newton method for finding a singularity of the nonsmooth vector fields. Basically, the new method combines a version of nonsmooth Newton method with a nonmonotone line search strategy. The global convergence analysis of the proposed method as well as results on its rate are established under mild assumptions. Finally, numerical experiments illustrating the practical advantages of the proposed scheme are reported.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:190:y:2021:i:1:d:10.1007_s10957-021-01881-4
    DOI: 10.1007/s10957-021-01881-4
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    References listed on IDEAS

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    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. 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. R. Behling & A. Fischer & M. Herrich & A. Iusem & Y. Ye, 2014. "A Levenberg-Marquardt method with approximate projections," Computational Optimization and Applications, Springer, vol. 59(1), pages 5-26, October.
    5. J. Y. Bello Cruz & O. P. Ferreira & L. F. Prudente, 2016. "On the global convergence of the inexact semi-smooth Newton method for absolute value equation," Computational Optimization and Applications, Springer, vol. 65(1), pages 93-108, September.
    6. Bittencourt, Tiberio & Ferreira, Orizon Pereira, 2015. "Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian manifolds," Applied Mathematics and Computation, Elsevier, vol. 261(C), pages 28-38.
    7. 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.
    8. 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.
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