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The gradient method for minimize a differentiable convex function onRiemannian manifolds with lower bounded sectional curvature is analyzed in thispaper. The analysis of the method is presented with three different finiteprocedures for determining the stepsize, namely, Lipschitz stepsize, adaptivestepsize and Armijo's stepsize. The first procedure requires that the objectivefunction has Lipschitz continuous gradient, which is not necessary for theother approaches. Convergence of the whole sequence to a minimizer, without anylevel set boundedness assumption, is proved. Iteration-complexity bound forfunctions with Lipschitz continuous gradient is also presented. Numericalexperiments are provided to illustrate the effectiveness of the method in thisnew setting and certify the obtained theoretical results. In particular, weconsider the problem of finding the Riemannian center of mass and the so-calledKarcher's mean. Our numerical experiences indicate that the adaptive stepsizeis a promising scheme that is worth considering.

Gradient Method for Optimization on Riemannian Manifolds with Lower Bounded Curvature