Riemannian conditional gradient methods for composite optimization problems

Abstract: In this paper, we propose Riemannian conditional gradient methods for minimizing composite functions, i.e., those that can be expressed as the sum of a smooth function and a geodesically convex one. We analyze the convergence of the proposed algorithms, utilizing three types of step-size strategies: adaptive, diminishing, and those based on the Armijo condition. We establish the convergence rate of (\mathcal{O}(1/k)) for the adaptive and diminishing step sizes, where (k) denotes the number of iterations. Additionally, we derive an iteration complexity of (\mathcal{O}(1/\epsilon^2)) for the Armijo step-size strategy to achieve (\epsilon)-optimality, where (\epsilon) is the optimality tolerance. Finally, the effectiveness of our algorithms is validated through some numerical experiments performed on the sphere and Stiefel manifolds.