Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields on Riemannian manifolds

TL;DR

This paper extends nonsmooth analysis to Riemannian manifolds and introduces a Newton method for finding singularities of locally Lipschitz vector fields, proving convergence and uniqueness under certain conditions.

Contribution

It develops a Newton method for nonsmooth vector fields on Riemannian manifolds and proves its convergence and uniqueness properties.

Findings

The Newton method converges locally to a singularity under mild conditions.

The method is well-defined and convergent for semismooth vector fields.

Convergence to a unique solution is established under Kantorovich assumptions.

Abstract

In this paper, we extend some results of nonsmooth analysis from Euclidean context to the Riemannian setting. In particular, we discuss the concept and some properties of locally Lipschitz continuous vector fields on Riemannian settings, such as Clarke generalized covariant derivative, upper semicontinuity and Rademacher theorem. We also present a version of Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields. Under mild conditions, we establish the well-definedness and local convergence of the sequence generated by the method in a neighborhood of a singularity. In particular, a local convergence result for semismooth vector fields is presented. Furthermore, under Kantorovich-type assumptions the convergence of the sequence generated by the Newton method to a solution is established, and its uniqueness in a suitable neighborhood of the starting point is verified.