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

Abstract

The “heavy ball with friction” dynamical system u¨+γu˙+∇Φ(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 u¨+γu˙+u−projC(u−μ∇Φ(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].