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We present a geometric multilevel optimization approach that smoothlyincorporates box constraints. Given a box constrained optimization problem, weconsider a hierarchy of models with varying discretization levels. Finer modelsare accurate but expensive to compute, while coarser models are less accuratebut cheaper to compute. When working at the fine level, multilevel optimisationcomputes the search direction based on a coarser model which speeds up updatesat the fine level. Moreover, exploiting geometry induced by the hierarchy thefeasibility of the updates is preserved. In particular, our approach extendsclassical components of multigrid methods like restriction and prolongation tothe Riemannian structure of our constraints.

Multi-level Geometric Optimization for Regularised Constrained Linear Inverse Problems