A note on minimization problems and multistep methods

We use the qualitative properties of the solution how of the gradient equation (x) over dot = -del f(x) to compute a local minimum of a real-valued function f. Under the regularity assumption of all equilibria we show a convergence result for bounded trajectories of a consistent, strictly stable linear multistep method applied to the gradient equation. Moreover, we compare the asymptotic features of the numerical and the exact solutions as done by Humphries, Stuart (1994) and Schropp (1995) for one-step methods. In the case of A(alpha)-stable formulae this leads to an efficient solver for stiff minimization problems.