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

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.