SPPID: Saddle-Point PID for Constrained Optimization

Abstract

This paper studies constrained optimization problems through the lens of feedback control. Building on the interpretation of Lagrange multipliers as feedback controllers, we propose the \emph{saddle-point PID (SPPID) dynamics}: a unified proportional--integral--derivative (PID) framework for continuous-time saddle-point dynamics. The proposed dynamics employ PID control for equality constraints and anti-windup PI control for inequality constraints. We show that SPPID is equivalent to a preconditioned primal--dual gradient flow of the augmented Lagrangian. This equivalence reveals the distinct role of each feedback component: integral action enforces constraint satisfaction, proportional action induces the augmented Lagrangian structure, and derivative action modifies the geometry of the primal dynamics via a state-dependent Riemannian metric. For convex problems, we establish convergence to the KKT set together with conditions for global asymptotic stability. For equality-constrained problems, we establish local exponential convergence for nonlinear equality constraints under standard assumptions, and show that projected gradient flow emerges as the infinite derivative-gain limit. For strongly convex problems with affine equality or inequality constraints, we establish global exponential convergence by leveraging contraction theory. Finally, we provide various numerical examples to illustrate the utility of SPPID.