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

• 成果类型: Article; Proceedings Paper
• 署名作者: Goudou, X.; Munier, J.
• 署名单位: Universite de Montpellier; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-007-0109-5
• 发表日期: 2009
• 页码: 173-191
• 关键词: proximal point algorithm; monotone-operators; hilbert-space; CONVERGENCE

摘要

We consider the gradient system x(t) + del Phi(x(t)) = 0 and the so-called heavy ball with friction dynamical system x(t) + lambda x(t) + del 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 Phi. 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 Phi is a Morse function, towards local minimum of Phi.