Multilevel Optimization: Geometric Coarse Models and Convergence Analysis

TL;DR

This paper introduces a multilevel optimization framework inspired by PDE multigrid methods, using coarse models for efficient descent directions and analyzing convergence under convexity assumptions, with promising numerical results.

Contribution

It proposes a novel multilevel approach for structured optimization problems that leverages coarse models and provides convergence analysis, extending to box constraints.

Findings

Rapid convergence far from the solution

Competitive performance with state-of-the-art methods

Effective for large-scale discrete tomography problems

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.