Metrically regular vector field and iterative processes for generalized equations in Hadamard manifolds. (English) Zbl 1387.90241
Summary: 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.
MSC:
| 90C30 | Nonlinear programming |
| 49M37 | Numerical methods based on nonlinear programming |
| 65K05 | Numerical mathematical programming methods |
Keywords:
generalized equation; metric regularity; proximal point method; Newton method; Zincenko’s method; superlinear convergence; Hadamard manifoldReferences:
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