  A variational formulation for accelerated optimization on normed vector
spaces was recently introduced in Wibisono et al., and later generalized to the
Riemannian manifold setting in Duruisseaux and Leok. This variational framework
was exploited on normed vector spaces in Duruisseaux et al. using time-adaptive
geometric integrators to design efficient explicit algorithms for symplectic
accelerated optimization, and it was observed that geometric discretizations
which respect the time-rescaling invariance and symplecticity of the Lagrangian
and Hamiltonian flows were substantially less prone to stability issues, and
were therefore more robust, reliable, and computationally efficient. As such,
it is natural to develop time-adaptive Hamiltonian variational integrators for
accelerated optimization on Riemannian manifolds. In this paper, we consider
the case of Riemannian manifolds embedded in a Euclidean space that can be
characterized as the level set of a submersion. We will explore how holonomic
constraints can be incorporated in discrete variational integrators to
constrain the numerical discretization of the Riemannian Hamiltonian system to
the Riemannian manifold, and we will test the performance of the resulting
algorithms by solving eigenvalue and Procrustes problems formulated as
optimization problems on the unit sphere and Stiefel manifold.
