Convergence Analysis of Gradient Algorithms on Riemannian Manifolds without Curvature Constraints and Application to Riemannian Mass
研究了在无曲率约束的黎曼流形上使用一般步长的梯度算法的收敛性，证明了局部凸性/拟凸性条件下的局部/全局收敛以及弱尖锐极小值下的线性收敛，并应用于黎曼L^p质量中心问题，改进了现有结果。
We study the convergence issue for the gradient algorithm (employing general step sizes) for optimization problems on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity (resp., weak sharp minima), local/global convergence (resp., linear convergence) results are established. As an application, the linear convergence properties of the gradient algorithm employing the constant step sizes and the Armijo step sizes for finding the Riemannian $L^p$ ($p\in[1,+\infty)$) centers of mass are explored, respectively, which in particular extend and/or improve the corresponding results in [B. Afsari, R. Tron, and R. Vidal, SIAM J. Control Optim., 51 (2013), pp. 2230--2260; G. C. Bento et al., J. Optim. Theory Appl., 183 (2019), pp. 977--992].