  In this paper we 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. In this scope 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.
