Sufficient Descent Riemannian Conjugate Gradient Method

This paper considers sufficient descent Riemannian conjugate gradient methods with line search algorithms. We propose two kinds of sufficient descent nonlinear conjugate gradient methods and prove these methods satisfy the sufficient descent condition even on Riemannian manifolds. One is the hybrid method combining the Fletcher-Reeves-type method with the Polak-Ribiere-Polyak-type method, and the other is the Hager-Zhang-type method, both of which are generalizations of those used in Euclidean space. Also, we generalize two kinds of line search algorithms that are widely used in Euclidean space. In addition, we numerically compare our generalized methods by solving several Riemannian optimization problems. The results show that the performance of the proposed hybrid method greatly depends regardless of the type of line search used. Meanwhile, the Hager-Zhang-type method has the fast convergence property regardless of the type of line search used.