The heavy ball with friction dynamical system for convex constrained minimization problems

Abstract

The “heavy ball with friction” dynamical system

$$ \ddot{u} + \gamma \dot{u} + \nabla \Phi (u) = 0$$

is a non-linear oscillator with damping (γ > 0). In [2], Alvarez proved that when H is a real Hilbert space and Ф : H → ℝ is a smooth convex function whose minimal value is achieved, then each trajectory t → u (t) of this system weakly converges towards a minimizer of Ф. We prove a similar result in the convex constrained case by considering the corresponding gradient-projection dynamical system

$$ \ddot{u} + \gamma \dot{u} + u - pro{{j}_{C}}(u - \mu \nabla \Phi (u)) = 0, $$

, where C is a closed convex subset of H. This result holds when H is a possibly infinite dimensional space, and extends, by using different technics, previous results by Antipin [1].

Partially supported by Comisión Nacional de Investigación Científica y Tecnológica de Chile under Fondecyt grant 1990884