Real-time Solution Strategy for Linearly Constrained Quadratic Programs with Proportional-Integral Control and Variants

We propose a generalized feedback system architecture to synthesize and analyze a class of continuous-time dynamics that solves linearly constrained quadratic programs (LCQPs) in real time. In particular, the architecture executes gradient descent on the cost function, invokes a deliberately engineered sector-bounded nonlinearity to penalize excursions in states away from the feasible space, and involves a family of linear controllers, including proportional-integral (PI) control and its variants to shape dynamic performance and minimize steady-state error. We establish the correspondence of the equilibria of the continuous-time dynamics with stationary points of the LCQP and provide numerical tests for stability leveraging the Kalman-Yakubovich-Popov lemma. We also comment on how specific variants of PI control present similarities to widely studied methods, including the augmented Lagrangian method, the primal-dual method, and gradient-flow dynamics for penalty program reformulations. Theoretical results are validated through simulations to benchmark performance and elucidate the design choices that emerge from the analysis.