A New Framework for Constrained Optimization via Feedback Control of Lagrange Multipliers

The continuous-time analysis of iterative algorithms for optimization has a long-standing history. This work introduces a novel framework for equality-constrained optimization based on control theory. The central concept is to design a feedback control system in which the Lagrange multipliers serve as the control inputs while the output represents the constraints. This system converges to a stationary point of the constrained optimization problem through suitable regulation. Concerning the Lagrange multipliers, we explore two control laws: proportional-integral control and feedback linearization. These choices lead to a variety of different methods. We rigorously develop the related algorithms, analyze their convergence theoretically, and present several numerical experiments that demonstrate their effectiveness compared to the state-of-the-art approaches.