On the globalization of Riemannian Newton method

Abstract

In the present paper, in order to find a singularity of a vector field defined on Riemannian manifolds, we present a new globalization strategy of Newton method and establish its global convergence with superlinear rate. In particular, this globalization generalizes for a general retraction the existing damped Newton's method. The presented global convergence analysis does not require any hypotesesis on singularity of the vector field. We applied the proposed method to solve the truncated singular value problem on the product of two Stiefel manifolds, the dextrous hand grasping problem on the cone of symmetric positive definite matrices and the Rayleigh quotient on the sphere. Moreover, some academic problems are solved. Numerical experiments are presented showing that the proposed algorithm has better robustness compared with the aforementioned method.