Open Problem: Two Riddles in Heavy-Ball Dynamics

This short paper presents two open problems on the widely used Polyak’s Heavy-Ball algorithm. The first problem is the method’s ability to exactly $\mathit{\text{accelerate}}$ in dimension one exactly. The second question regards the behavior of the method for parameters for which it seems that neither a Lyapunov nor a cycle exists. For both problems, we provide a detailed description of the problem and elements of an answer.