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.

翻译：我们提出一种几何多层级优化方法，能够平滑地融入箱型约束。针对箱型约束优化问题，我们构建了一个包含不同离散化层级的模型层级结构。精细模型精度高但计算代价大，而粗糙模型精度较低但计算代价小。在精细层级求解时，多层级优化基于较粗糙模型计算搜索方向，从而加速精细层级的更新过程。此外，通过利用层级结构诱导的几何特性，更新过程的可行性得以保持。特别地，我们的方法将多重网格方法中的经典组件（如限制与延拓）扩展至约束条件的黎曼几何结构。