  We present a geometric multilevel optimization approach that smoothly
incorporates box constraints. Given a box constrained optimization problem, we
consider a hierarchy of models with varying discretization levels. Finer models
are accurate but expensive to compute, while coarser models are less accurate
but cheaper to compute. When working at the fine level, multilevel optimisation
computes the search direction based on a coarser model which speeds up updates
at the fine level. Moreover, exploiting geometry induced by the hierarchy the
feasibility of the updates is preserved. In particular, our approach extends
classical components of multigrid methods like restriction and prolongation to
the Riemannian structure of our constraints.
