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

Orizon P. Ferreira (), Célia Jean-Alexis () and Alain Piétrus ()
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Orizon P. Ferreira: Universidade Federal de Goiás
Célia Jean-Alexis: Université des Antilles
Alain Piétrus: Université des Antilles

Journal of Optimization Theory and Applications, 2017, vol. 175, issue 3, No 2, 624-651

Abstract: 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.

Keywords: Generalized equation; Metric regularity; Proximal point method; Newton method; Zincenko’s method; Superlinear convergence; Hadamard manifold; 90C30; 49M37; 65K05 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10957-017-1195-z

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