Multilevel Optimization: Geometric Coarse Models and Convergence Analysis
Ferdinand Vanmaele, Yara Elshiaty, Stefania Petra
TL;DR
The paper develops a geometric multilevel framework for convex optimization with a grid hierarchy, constructing a coarse model that provides descent directions on the fine grid without requiring Hessians on the fine level. Through a two-grid cycle and a coarse correction strategy, the authors prove linear convergence under $μ$-strong convexity and $L$-Lipschitz continuity, with a finite number of coarse corrections in coercive settings. They extend the method to box-constrained problems and demonstrate its practicality via large-scale discrete tomography experiments, where the geometric coarse models yield rapid early convergence and competitive performance against L-BFGS-B. The work highlights computational savings from geometry-based coarse models and their applicability to imaging problems, while noting that near the solution superlinear methods may prevail and outlining avenues for future work on active-set handling and extensions to nonsmooth cases.
Abstract
We study multilevel techniques, commonly used in PDE multigrid literature, to solve structured optimization problems. For a given hierarchy of levels, we formulate a coarse model that approximates the problem at each level and provides a descent direction for the fine-grid objective using fewer variables. Unlike common algebraic approaches, we assume the objective function and its gradient can be evaluated at each level. Under the assumptions of strong convexity and gradient L-smoothness, we analyze convergence and extend the method to box-constrained optimization. Large-scale numerical experiments on a discrete tomography problem show that the multilevel approach converges rapidly when far from the solution and performs competitively with state-of-the-art methods.