Contractivity of Runge-Kutta methods for convex gradient systems

TL;DR

This paper investigates whether Runge-Kutta methods preserve the contractive property of solutions in convex gradient systems, revealing that some schemes fail to do so regardless of step size, challenging previous assumptions.

Contribution

The authors construct explicit examples showing certain Runge-Kutta schemes do not maintain contractivity for any step size, despite prior results suggesting conditions for contractive behavior.

Findings

Explicit Runge-Kutta schemes can fail to be contractive regardless of step size.

Euler's method is optimal for contractivity among explicit schemes.

Constructed convex potential demonstrates failure of contractivity in some schemes.

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 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 $h$.