A new framework for constrained optimization via feedback control of Lagrange multipliers

TL;DR

This paper introduces a novel continuous-time control framework for equality-constrained optimization, leveraging feedback control of Lagrange multipliers to ensure convergence to stationary points.

Contribution

It develops a new control-theoretic approach for constrained optimization, analyzing different feedback laws and providing rigorous convergence proofs and numerical validation.

Findings

The proposed methods converge to stationary points.

Feedback control laws improve convergence stability.

Numerical experiments demonstrate effectiveness over existing methods.

Abstract

The continuous-time analysis of existing iterative algorithms for optimization has a long history. This work proposes a novel continuous-time control-theoretic framework for equality-constrained optimization. The key idea is to design a feedback control system where the Lagrange multipliers are the control input, and the output represents the constraints. The system converges to a stationary point of the constrained optimization problem through suitable regulation. Regarding the Lagrange multipliers, we consider two control laws: proportional-integral control and feedback linearization. These choices give rise to a family of different methods. We rigorously develop the related algorithms, theoretically analyze their convergence and present several numerical experiments to support their effectiveness concerning the state-of-the-art approaches.