Hybrid Heavy-Ball Systems: Reset Methods for Optimization with Uncertainty

Momentum methods for convex optimization often rely on precise choices of algorithmic parameters, based on knowledge of problem parameters, in order to achieve fast convergence, as well as to prevent oscillations that could severely restrict applications of these algorithms to cyber-physical systems. To address these issues, we propose two dynamical systems, named the Hybrid Heavy-Ball System and Hybrid-inspired Heavy-Ball System, which employ a feedback mechanism for driving the momentum state toward zero whenever it points in undesired directions. We describe the relationship between the proposed systems and their discrete-time counterparts, deriving conditions based on linear matrix inequalities for ensuring exponential rates in both continuous time and discrete time. We provide numerical LMI results to illustrate the effects of our reset mechanisms on convergence rates in a setting that simulates uncertainty of problem parameters. Finally, we numerically demonstrate the efficiency and avoidance of oscillations of the proposed systems when solving both strongly convex and non-strongly convex problems.