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 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. We prove that there are RK schemes that for arbitrarily small $h$ do not behave contractively. We derive a sufficient condition that guarantees the contractivity of a given RK method for a given $h$. It turns out that there are explicit RK schemes that behave contractively whenever $Lh$ is below a scheme-dependent constant, which is some cases coincides with the one given by linear stability analysis. We show that, among all consistent, explicit RK methods, Euler's rule is optimal in this regard.

中文翻译：

---

Runge的收缩性--凸梯度系统的Kutta方法

我们考虑将 Runge-Kutta (RK) 方法应用于梯度系统 $(d/dt)x = -\nabla V(x)$，其中，与许多优化问题一样，$V$ 是凸的，而 $\nabla V $（全局）Lipschitz-连续与Lipschitz 常数$L$。该系统的解具有收缩性，即两个解 $x(t)$ 和 $\widetilde{x}(t)$ 之间的距离是 $t$ 的非增函数。然后，研究离散解是否会发生类似的收缩（至少对于适当的小步长 $h$）是很有趣的。我们证明存在 RK 方案对于任意小的 $h$ 不具有契约行为。我们推导出一个充分条件来保证给定的 RK 方法对于给定的 $h$ 的收缩性。事实证明，只要 $Lh$ 低于与方案相关的常数，就会有显式的 RK 方案表现得收缩，这在某些情况下与线性稳定性分析给出的情况一致。我们表明，在所有一致的显式 RK 方法中，欧拉规则在这方面是最优的。