# A Riemannian Derivative-Free Polak-Ribiere-Polyak Method for Tangent Vector Field

## TL;DR

This paper introduces a derivative-free optimization method on Riemannian manifolds for finding zeros of tangent vector fields, combining a Polak-Ribiere-Polyak approach with a hybrid Newton method, supported by convergence analysis and numerical experiments.

## Contribution

It proposes a novel Riemannian derivative-free optimization algorithm with a hybrid scheme and convergence guarantees, addressing tangent vector field zero-finding problems.

## Key findings

- The method converges globally under mild conditions.
- Numerical experiments demonstrate improved efficiency.
- The hybrid approach outperforms standalone methods.

## Abstract

This paper is concerned with the problem of finding a zero of a tangent vector field on a Riemannian manifold. We first reformulate the problem as an equivalent Riemannian optimization problem. Then we propose a Riemannian derivative-free Polak-Ribi\'ere-Polyak method for solving the Riemannian optimization problem, where a non-monotone line search is employed. The global convergence of the proposed method is established under some mild assumptions. To further improve the efficiency, we also provide a hybrid method, which combines the proposed geometric method with the Riemannian Newton method. Finally, some numerical experiments are reported to illustrate the efficiency of the proposed method.