Multilevel Bregman Proximal Gradient Descent

TL;DR

The paper introduces ML BPGD, a multilevel optimization method for constrained convex problems with Bregman geometries, proving linear convergence and demonstrating effectiveness in image reconstruction.

Contribution

It extends multilevel optimization to Bregman geometries and constrained problems, providing theoretical convergence guarantees and practical validation.

Findings

Proves global linear convergence rate for ML BPGD.

Validates effectiveness in image reconstruction tasks.

Provides theoretical guarantees for multilevel Bregman framework.

Abstract

We present the Multilevel Bregman Proximal Gradient Descent (ML BPGD) method, a novel multilevel optimization framework tailored to constrained convex problems with relative Lipschitz smoothness. Our approach extends the classical multilevel optimization framework (MGOPT) to handle Bregman-based geometries and constrained domains. We provide a rigorous analysis of ML BPGD for multiple coarse levels and establish a global linear convergence rate. We demonstrate the effectiveness of ML BPGD in the context of image reconstruction, providing theoretical guarantees for the well-posedness of the multilevel framework and validating its performance through numerical experiments.