On Weakly Contracting Dynamics for Convex Optimization

We analyze the convergence behavior of \emph{globally weakly} and \emph{locally strongly contracting} dynamics. Such dynamics naturally arise in the context of convex optimization problems with a unique minimizer. We show that convergence to the equilibrium is \emph{linear-exponential}, in the sense that the distance between each solution and the equilibrium is upper bounded by a function that first decreases linearly and then exponentially. As we show, the linear-exponential dependency arises naturally in certain dynamics with saturations. Additionally, we provide a sufficient condition for local input-to-state stability. Finally, we illustrate our results on, and propose a conjecture for, continuous-time dynamical systems solving linear programs.