Gradient Method for Optimization on Riemannian Manifolds with Lower Bounded Curvature

The gradient method for minimizing a differentiable convex function on Riemannian manifolds with lower bounded sectional curvature is analyzed in this paper. An analysis of the method with three different finite procedures for determining the step size (namely, Lipschitz step size, adaptive step size, and Armijo's step size) is presented. The first procedure requires that the objective function has Lipschitz continuous gradient, which is not necessary for the other approaches. Convergence of the whole sequence to a minimizer, without any level set boundedness assumption, is proved. The iteration-complexity bound for functions with Lipschitz continuous gradient is also presented. Numerical experiments are provided to illustrate the effectiveness of the method in this new setting and certify the theoretical results. In particular, we consider the problem of finding the Riemannian center of mass and the so-called Karcher mean. Our numerical experiences indicate that the adaptive step size is a promising scheme that is worth considering.