The heavy ball with friction method, I. the continuous dynamical system: global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system

摘要

Let H be a real Hilbert space and Φ: H → R a continuously differentiable function, whose gradient is Lipschitz continuous on bounded sets. We study the nonlinear dissipative dynamical system: x(t) + λx(t) + ▽Φ(x(t)) = 0, λ > 0, plus Cauchy data, mainly in view of the unconstrained minimization of the function Φ. New results concerning the convergence of a solution to a critical point are given in various situations , including when Φ is convex (possibly with multiple minima) or is a Morse function ( the critical point being then generically a local minimum); a counterexample shows that, without peculiar assumptions, a trajectory may not converge. By following the trajectories, we obtain a method for exploring local minima of Φ. A singular perturbation analysis links our results with those concerning gradient systems.