An efficient damped Newton-type algorithm with globalization strategy on Riemannian manifolds

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We propose a new globalization strategy of the damped Newton method for finding singularities of a vector field on Riemannian manifolds. We establish its global convergence with a superlinear rate. In particular, this globalization generalizes the known damped Newton’s method for general retraction. The global convergence analysis presented here does not require any hypothesis regarding a 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 and an academic problem on the cone of symmetric positive definite matrices, the Rayleigh quotient and a problem of finding zeros of a non-conservative vector field on the sphere. Numerical experiments are presented, showing that the proposed algorithm is more robust than the known damped Newton’s method.