This paper considers sufficient descent Riemannian conjugate gradient methods with line search algorithms. We propose two kinds of sufficient descent nonlinear conjugate gradient method and prove that these methods satisfy the sufficient descent condition on Riemannian manifolds. One is a hybrid method combining a Fletcher–Reeves-type method with a Polak–Ribière–Polyak-type method, and the other is a Hager–Zhang-type method, both of which are generalizations of those used in Euclidean space. Moreover, we prove that the hybrid method has a global convergence property under the strong Wolfe conditions and the Hager–Zhang-type method has the sufficient descent property regardless of whether a line search is used or not. Further, we review two kinds of line search algorithm on Riemannian manifolds and numerically compare our generalized methods by solving several Riemannian optimization problems. The results show that the performance of the proposed hybrid methods greatly depends on 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.

本文考虑了具有线搜索算法的充分下降黎曼共轭梯度方法。我们提出了两种充分下降非线性共轭梯度方法，并证明这些方法满足黎曼流形上的充分下降条件。一种是将 Fletcher-Reeves 型方法与 Polak-Ribière-Polyak 型方法相结合的混合方法，另一种是 Hager-Zhang 型方法，两者都是欧几里德空间中使用的方法的推广。此外，我们证明了混合方法在强沃尔夫条件下具有全局收敛性，而无论是否使用线搜索，Hager-Zhang 型方法都具有足够的下降性。更多，我们回顾了黎曼流形上的两种线搜索算法，并通过解决几个黎曼优化问题对我们的广义方法进行了数值比较。结果表明，所提出的混合方法的性能在很大程度上取决于所使用的线搜索类型。同时，无论使用哪种线搜索，Hager-Zhang 型方法都具有快速收敛的特性。