We introduce an autonomous system with closed-loop damping for first-order convex optimization. While, to this day, optimal rates of convergence are only achieved by non-autonomous methods via open-loop damping (e.g., Nesterov's algorithm), we show that our system is the first one featuring a closed-loop damping while exhibiting a rate arbitrarily close to the optimal one. We do so by coupling the damping and the speed of convergence of the system via a well-chosen Lyapunov function. We then derive a practical first-order algorithm called LYDIA by discretizing our system, and present numerical experiments supporting our theoretical findings.

翻译：我们提出了一种具有闭环阻尼的一阶凸优化自治系统。目前，最优收敛速率仅能通过非自治方法（如内斯特罗夫算法）经由开环阻尼实现，而本文证明所提系统是首个在具备闭环阻尼的同时达到任意接近最优速率的系统。通过选取恰当的李雅普诺夫函数，我们将系统阻尼与收敛速度进行耦合。随后通过对系统进行离散化，推导出名为LYDIA的实用一阶算法，并通过数值实验验证了理论结果。