Exponential convergence of primal-dual dynamics for multi-block problems under local error bound condition

We study composite optimization problems in which the objective function is given by the sum of a smooth strongly convex term and multiple, potentially non-differentiable, convex regularizers. For a class of problems that satisfy a structural property expressed in terms of a local error bound condition, we establish the existence of a finite time after which a primal-dual method based on the proximal augmented Lagrangian converges exponentially fast to the set of optimal primal-dual variables.