Projected Gradient Method on Hadamard Manifolds

Abstract

We study constrained smooth optimization problems on Hadamard manifolds with closed geodesically convex feasible sets. We analyze two projected gradient schemes: one with a constant stepsize and another with a backtracking line search. The constant-stepsize scheme is analyzed under the assumption that the objective function has a Lipschitz continuous Riemannian gradient, whereas the backtracking variant does not require this assumption to establish stationarity of accumulation points. For both schemes, we prove that every accumulation point of the generated sequence is first-order stationary under the respective assumptions, without requiring compactness of the feasible set; compactness is needed only to ensure the existence of accumulation points. When the objective function has a Lipschitz continuous Riemannian gradient, we derive iteration-complexity bounds of order \(O(1/\sqrt{N})\) for projection-based stationarity measures for both schemes, together with the corresponding \(\varepsilon\)-complexity estimates. For the backtracking scheme, the complexity analysis additionally requires the trial line-search stepsizes to be uniformly bounded away from zero. Under the same respective assumptions, the generated sequences are also asymptotically regular. Finally, we illustrate the practical performance of the methods by solving constrained Karcher mean problems on the manifold of symmetric positive definite matrices.