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A variational formulation of accelerated optimization on normed spaces wasrecently introduced by considering a specific family of time-dependent BregmanLagrangian and Hamiltonian systems whose corresponding trajectories converge tothe minimizer of the given convex function at an arbitrary accelerated rate of$\mathcal{O}(1/t^p)$. This framework has been exploited using time-adaptivegeometric integrators to design efficient explicit algorithms for symplecticaccelerated optimization. It was observed that geometric discretizations weresubstantially less prone to stability issues, and were therefore more robust,reliable, and computationally efficient. More recently, this variationalframework has been extended to the Riemannian manifold setting by considering amore general family of time-dependent Bregman Lagrangian and Hamiltoniansystems on Riemannian manifolds. It is thus natural to develop time-adaptiveHamiltonian variational integrators for accelerated optimization on Riemannianmanifolds. In the past, Hamiltonian variational integrators have beenconstructed with holonomic constraints, but the resulting algorithms wereimplicit in nature, which significantly increased their cost per iteration. Inthis paper, we will test the performance of explicit methods based onHamiltonian variational integrators combined with projections that constrainthe numerical solution to remain on the constraint manifold.

Accelerated Optimization on Riemannian Manifolds via Projected Variational Integrators