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

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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.

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Acknowledgements

This work was completed while the first author was visiting the second and third authors at the Laboratoire de Mathématiques Informatique et Applications (LAMIA) of Université des Antilles (UA) during the period February to March 2017. He is grateful to their host institution for the congenial scientific atmosphere that it has provided during his visit. The work was financially supported by FAPEG and CNPq Grants: 305158/2014-7 and 302473/2017-3.

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Authors and Affiliations

  1. IME, Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    Orizon P. Ferreira

  2. Laboratoire LAMIA, Département de Mathématiques et Informatique, EA4540, Université des Antilles, Campus de Fouillole, 97159, Pointe-à-Pitre, France

    Célia Jean-Alexis & Alain Piétrus

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  1. Orizon P. Ferreira
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  2. Célia Jean-Alexis
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  3. Alain Piétrus
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Correspondence to Orizon P. Ferreira.

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Communicated by Sándor Zoltán Németh.

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Ferreira, O.P., Jean-Alexis, C. & Piétrus, A. Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds. J Optim Theory Appl 175, 624–651 (2017). https://doi.org/10.1007/s10957-017-1195-z

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