  A variational framework for accelerated optimization was recently introduced
on normed vector spaces and Riemannian manifolds in Wibisono et al. (2016) and
Duruisseaux and Leok (2021). It was observed that a careful combination of
timeadaptivity and symplecticity in the numerical integration can result in a
significant gain in computational efficiency. It is however well known that
symplectic integrators lose their near energy preservation properties when
variable time-steps are used. The most common approach to circumvent this
problem involves the Poincare transformation on the Hamiltonian side, and was
used in Duruisseaux et al. (2021) to construct efficient explicit algorithms
for symplectic accelerated optimization. However, the current formulations of
Hamiltonian variational integrators do not make intrinsic sense on more general
spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian
variational integrators are well-defined on manifolds, so we develop here a
framework for time-adaptivity in Lagrangian variational integrators and use the
resulting geometric integrators to solve optimization problems on normed vector
spaces and Lie groups.
