A Global Version of the Newton Method for Finding a Singularity of the Nonsmooth Vector Fields on Riemannian Manifolds

TL;DR

This paper introduces a global Newton method with a nonmonotone line search for finding singularities in nonsmooth vector fields on Riemannian manifolds, supported by numerical experiments demonstrating its effectiveness.

Contribution

It develops a globally convergent Newton method with a nonmonotone line search for nonsmooth vector fields on Riemannian manifolds, extending previous local methods.

Findings

Numerical experiments show improved convergence and practical performance.

The proposed method effectively finds singularities in complex nonsmooth vector fields.

The approach enhances existing algorithms with global convergence guarantees.

Abstract

This paper is concerned with an algorithm for finding a singularity of the nonsmooth vector fields. Firstly, we discuss the main results of the Newton method presented in [1] for solving the aforementioned problem. Combining this method with a nonmonotone line search strategy, we then propose a global version of the Newton Method. Finally, numerical experiments illustrating the practical advantages of the proposed scheme are reported.