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

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