Contractivity of Runge-Kutta methods for convex gradient systems

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 widetildex(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.