Global Convergence and Asymptotic Optimality of the Heavy Ball Method for a Class of Nonconvex Optimization Problems

In this letter we revisit the famous heavy ball method and study its global convergence for a class of non-convex problems with sector-bounded gradient. We characterize the parameters that render the method globally convergent and yield the best
$R$
-convergence factor. We show that for this family of functions, this convergence factor is superior to the factor obtained from the triple momentum method.