Contractivity of Runge--Kutta Methods for Convex Gradient Systems

Abstract

We consider the application of Runge--Kutta (RK) methods to gradient systems $(d/dt)x = -\nabla V(x)$, where, as in many optimization problems, $V$ is convex and $\nabla V$ (globally) Lipschitz-continuous with Lipschitz constant $L$. Solutions of this system behave contractively, i.e., the Euclidean distance between two solutions $x(t)$ and $\widetilde{x}(t)$ is a nonincreasing function of $t$. It is then of interest to investigate whether a similar contraction takes place, at least for suitably small step sizes $h$, for the discrete solution. Dahlquist and Jeltsch's results imply that (1) there are explicit RK schemes that behave contractively whenever $Lh$ is below a scheme-dependent constant and (2) Euler's rule is optimal in this regard. We prove, however, by explicit construction of a convex potential using ideas from robust control theory, that there exist RK schemes that fail to behave contractively for any choice of the time-step $h$.