  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.
