  A variational formulation for accelerated optimization on normed spaces was
introduced in Wibisono et al., and later generalized to Riemannian manifolds in
Duruisseaux and Leok. This variational framework was exploited on normed 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.
