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We consider the minimization of a cost function $f$ on a manifold $M$ usingRiemannian gradient descent and Riemannian trust regions (RTR). We focus onsatisfying necessary optimality conditions within a tolerance $\varepsilon$.Specifically, we show that, under Lipschitz-type assumptions on the pullbacksof $f$ to the tangent spaces of $M$, both of these algorithms produce pointswith Riemannian gradient smaller than $\varepsilon$ in $O(1/\varepsilon^2)$iterations. Furthermore, RTR returns a point where also the RiemannianHessian's least eigenvalue is larger than $-\varepsilon$ in$O(1/\varepsilon^3)$ iterations. There are no assumptions on initialization.The rates match their (sharp) unconstrained counterparts as a function of theaccuracy $\varepsilon$ (up to constants) and hence are sharp in that sense.  These are the first deterministic results for global rates of convergence toapproximate first- and second-order Karush-Kuhn-Tucker points on manifolds.They apply in particular for optimization constrained to compact submanifoldsof $\mathbb{R}^n$, under simpler assumptions.

Global rates of convergence for nonconvex optimization on manifolds