切向量场的黎曼无导数 Polak-Ribiere-Polyak 方法
A Riemannian Derivative-Free Polak-Ribiere-Polyak Method for Tangent Vector Field
AI总结：
本文研究 Riemannian 流形上切向量场求零点问题，将其转化为等价优化问题，提出采用非单调线搜索的无导数 Polak-Ribiére-Polyak 方法，证明全局收敛，并结合 Riemannian Newton 方法提升效率，数值实验验证了方法的有效性。
摘要：
本文研究在 Riemannian 流形上寻找切向量场零点的问题。我们首先将该问题转化为等价的 Riemannian 优化问题。随后，我们提出一种 Riemannian derivative-free Polak-Ribiére-Polyak 方法求解该优化问题，其中采用了非单调线搜索。在温和假设下证明了所提方法的全局收敛性。为进一步提升效率，我们还提供一种混合方法，将所提几何方法与 Riemannian Newton 方法相结合。最后，报告了数值实验以说明所提方法的效率。
英文摘要：
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-Ribiére-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.
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