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Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds

Author

Listed:
  • Orizon P. Ferreira

    (Universidade Federal de Goiás)

  • Célia Jean-Alexis

    (Université des Antilles)

  • Alain Piétrus

    (Université des Antilles)

Abstract

This paper is focused on the problem of finding a singularity of the sum of two vector fields defined on a Hadamard manifold, or more precisely, the study of a generalized equation in a Riemannian setting. We extend the concept of metric regularity to the Riemannian setting and investigate its relationship with the generalized equation in this new context. In particular, a version of Graves’s theorem is presented and we also define some concepts related to metric regularity, including the Aubin property and the strong metric regularity of set-valued vector fields. A conceptual method for finding a singularity of the sum of two vector fields is also considered. This method has as particular instances: the proximal point method, Newton’s method, and Zincenko’s method on Hadamard manifolds. Under the assumption of metric regularity at the singularity, we establish that the methods are well defined in a suitable neighborhood of the singularity. Moreover, we also show that each sequence generated by these methods converges to this singularity at a superlinear rate.

Suggested Citation

  • Orizon P. Ferreira & Célia Jean-Alexis & Alain Piétrus, 2017. "Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 624-651, December.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:3:d:10.1007_s10957-017-1195-z
    DOI: 10.1007/s10957-017-1195-z
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    References listed on IDEAS

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    1. 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.
    2. J. H. Wang & G. López & V. Martín-Márquez & C. Li, 2010. "Monotone and Accretive Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 691-708, September.
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