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arXiv · 2403.07572

On Weakly Contracting Dynamics for Convex Optimization

Veronica Centorrino·Alexander Davydov·Anand Gokhale·Giovanni Russo·Francesco Bullo
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Abstract

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.

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Veronica Centorrino, Alexander Davydov, Anand Gokhale, Giovanni Russo, Francesco Bullo. 2024-03-12. On Weakly Contracting Dynamics for Convex Optimization. https://arxiv.org/abs/2403.07572

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