Near-optimal Closed-loop Method via Lyapunov Damping for Convex Optimization

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.

@article{MCO26,
  title        = {Near-optimal Closed-loop Method via Lyapunov Damping for Convex Optimization},
  author       = {S. Maier and C. Castera and P. Ochs},
  year         = {2026},
  journal      = {Journal of Optimization Theory and Applications},
  number       = {93},
  volume       = {208}
}