A Riemannian BFGS Method Without Differentiated Retraction for Nonconvex Optimization Problems
提出一种基于谨慎更新和弱线搜索条件的黎曼BFGS方法，证明其在非凸目标函数下全局收敛到稳定点且超线性收敛到非退化极小点，无需收缩的微分信息。
In this paper, a Riemannian BFGS method for minimizing a smooth function on a Riemannian manifold is defined, based on a Riemannian generalization of a cautious update and a weak line search condition. It is proven that the Riemannian BFGS method converges (i) globally to stationary points without assuming the objective function to be convex and (ii) superlinearly to a nondegenerate minimizer. Using the weak line search condition removes the need for information from differentiated retraction. The joint matrix diagonalization problem is chosen to demonstrate the performance of the algorithms with various parameters, line search conditions, and pairs of retraction and vector transport. A preliminary version can be found in [Numerical Mathematics and Advanced Applications: ENUMATH 2015, Lect. Notes Comput. Sci. Eng. 112, Springer, New York, 2016, pp. 627--634].