The gradient and heavy ball with friction dynamical systems: the quasiconvex case

Abstract

We consider the gradient system $$\dot x(t)+\nabla\phi(x(t))=0$$ and the so-called heavy ball with friction dynamical system $$\ddot x(t) +\lambda\dot x(t)+\nabla\phi(x(t))=0$$, as well as an implicit discrete (proximal) version of it, and study the asymptotic behavior of their solutions in the case of a smooth and quasiconvex objective function Φ. Minimization properties of trajectories are obtained under various additional assumptions. We finally show a minimizing property of the heavy ball method which is not shared by the gradient method: the genericity of the convergence of each trajectory, at least when Φ is a Morse function, towards local minimum of Φ.