Convergence Analysis of Gradient Algorithms on Riemannian Manifolds without Curvature Constraints and Application to Riemannian Mass

We study the convergence issue for gradient algorithm (employing general step sizes) optimization problems on Riemannian manifolds (without curvature constraints). Under assumption of local convexity/quasi-convexity (resp., weak sharp minima), local/global linear convergence) results are established. As an application, properties employing constant sizes and Armijo finding $L^p$ ($p\in[1,+\infty)$) centers mass explored, respectively, which in particular extend and/or improve corresponding [B. Afsari, R. Tron, Vidal, SIAM J. Control Optim., 51 (2013), pp. 2230--2260; G. C. Bento et al., Optim. Theory Appl., 183 (2019), 977--992].