Using dynamical systems methods to solve minimization problems

摘要

One possibility to compute a local minimum of a real-valued function f of N unknowns is to solve the gradient differential equation x = - ▽f(x). In the present paper we derive a convergence result for minimization problems by discretizing this equation via fixed time-stepping one-step methods. We compare the asymptotic features of the numerical and the exact solutions. Furthermore, we show that for a certain class of one-step methods the totality of the discrete and the continuous ω-limit sets coincide if the stepsize is sufficiently small and if all equilibria of the gradient differential equation are regular.