In this talk, we first propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. To this end we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal
ADMM algorithm.

In the second part of the talk we give an outlook on a second order primal dual dynamical system with asymptotic vanishing term and on its fast convergence properties.

The talk relies on the papers (Bot ̧, Csetnek, La’szlo’, JDE, 2020) and (Bot ̧Nguyen, JDE, 2021).

Zoom meeting: Link