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 $\tilde{x}(t)$ 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 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 exists RK schemes that fail to behave contractively for any choice of the time-step $h$.

13 pages, 2 figures