Continuous-Time Heavy-Ball Gradient Method: Safety, Stability and Robustness

This letter proposes a continuous-time heavy-ball dynamical system to solve a constrained nonlinear optimization problem, ensuring forward invariance of the constraint set. Recent research in this area has focused on anytime solution for the nonlinear optimization problem, based on first-order gradient dynamics. Unlike the first-order gradient flows, the heavy-ball dynamics — being second-order — calls for the use of notions from exponential control barrier functions to ensure the safety of the closed-loop system with respect to the constraint set. The proposed heavy-ball dynamics is motivated by its added low-pass filtering capability to noisy gradients, which arise in stochastic gradient methods. Furthermore, we pose the problem of adversarial inputs or uncertainties in the constraint functions as a robust optimization problem and ensure the safety of the nominal constraint set. The results are validated on the extended Fermat-Torricelli problem, with non-convex constraint sets.