A Dynamical System Associated with Newton's Method for Parametric Approximations of Convex Minimization Problems

摘要

We study the existence and asymptotic convergence when t → +∞ for the trajectories generated by ▽~2 f(u(t), ε(t))u(t) + ε(t)((partial deriv)~2f)/((partial deriv)ε(partial deriv)x)(u(t), ε(t)) + ▽f(u(t), ε(t)) = 0, where {f(·,ε)}_(ε > 0) is a parametric family of convex functions which approximates a given convex function f we want to minimize, and ε(t) is a parametrization such that ε(t) → 0 when t → +∞. This method is obtained from the following variational characterization of Newton's method: (P_t~ε) u(t) ∈ Argmin{f(x, ε(t)) - e~(-1)<▽f(u_0, ε_0), x>:x ∈ H}, where H is a real Hilbert space. We find conditions on the approximating family f(·, ε) and the parametrization ε(t) to ensure the norm convergence of the solution trajectories u(t) toward a particular minimizer of f. The asymptotic estimates obtained allow us to study the rate of convergence as well. The results are illustrated through some applications to barrier and penalty methods for linear programming, and to viscosity methods for an abstract noncoercive variational problem. Comparisons with the steepest descent method are also provided.