# Contractivity of Runge-Kutta methods for convex gradient systems

arXiv: https://arxiv.org/abs/1909.09971 · HTML: https://mlsanity.com/paper/1909.09971/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Sep 2019.

## Built on by (most cited first, 4 of 4)

- [Designing Stable Neural Networks using Convex Analysis and ODEs](https://mlsanity.com/paper/2306.17332/) (2023)
- [Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations](https://mlsanity.com/paper/2104.12384/) (2021)
- [Revisiting the Role of Euler Numerical Integration on Acceleration and Stability in Convex Optimization](https://mlsanity.com/paper/2102.11537/) (2021)
- [Randomised Splitting Methods and Stochastic Gradient Descent](https://mlsanity.com/paper/2504.04274/) (2025)

## Abstract (arXiv)

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.