Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the optimization model, and as a result, they outperform single-level methods, especially for large-scale models. The impressive performance of multilevel optimization methods is an empirical observation, and no theoretical explanation has so far been proposed. In order to address this issue, we study the convergence properties of a multilevel method that is motivated by second-order methods. We take the first step toward establishing how the structure of an optimization problem is related to the convergence rate of multilevel algorithms.

受线性方程组的多重网格方法的启发，提出了多级优化方法来解决结构化优化问题。多级方法对优化模型的结构做出了更多假设，因此，它们优于单级方法，尤其是对于大型模型。多级优化方法令人印象深刻的性能是一种经验观察，到目前为止还没有提出任何理论解释。为了解决这个问题，我们研究了由二阶方法驱动的多级方法的收敛特性。我们迈出了确定优化问题的结构如何与多级算法的收敛速度相关的第一步。