Improved convergence rates and trajectory convergence for primal-dual dynamical systems with vanishing damping

TL;DR

This paper introduces a novel second-order primal-dual dynamical system with vanishing damping that guarantees convergence in convex optimization problems, improving understanding of trajectory behavior and linking to Nesterov's method.

Contribution

It provides the first convergence guarantees for primal-dual trajectories with vanishing damping and connects these results to classical accelerated gradient methods.

Findings

Fast convergence of primal-dual gap, feasibility, and objective value.

Weak convergence of trajectories to primal-dual optimal solutions.

Reveals connections to Nesterov's accelerated gradient method.

Abstract

In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with asymptotically vanishing damping term. The system is formulated in terms of the augmented Lagrangian associated to the minimization problem. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. In case the objective function has Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically weakly converges to a primal-dual optimal solution of the underlying minimization problem. To the best of our knowledge, this is the first result which guarantees the convergence of the trajectory generated by a primal-dual dynamical system with asymptotic vanishing damping. Moreover, we will rediscover in case of the unconstrained minimization of a convex differentiable function with Lipschitz continuous gradient all convergence statements obtained in the literature for Nesterov's accelerated gradient method.