Feedback control of Lagrange multipliers for non-smooth constrained optimization

TL;DR

This paper introduces a control-theoretic approach to solve non-smooth constrained optimization problems by designing controllers that ensure convergence to stationary points, with theoretical guarantees and numerical validation.

Contribution

It develops a novel control framework based on the proximal augmented Lagrangian, leading to two new algorithms with proven exponential convergence.

Findings

The proposed methods achieve global exponential convergence under strong convexity.

Numerical experiments show competitive performance against existing approaches.

Abstract

In this work, we develop a control-theoretic framework for constrained optimization problems with composite objective functions including non-differentiable terms. Building on the proximal augmented Lagrangian formulation, we construct a plant whose equilibria correspond to the stationary points of the optimization problem. Within this framework, we propose two control strategies - a static controller and a dynamic controller - leading to two novel optimization algorithms. We provide a theoretical analysis, establishing global exponential convergence under strong convexity assumptions. Finally, we demonstrate the effectiveness of the proposed methods through numerical experiments, benchmarking their performance against state-of-the-art approaches.