On the asymptotic stability of proximal algorithms for convex optimization problems with multiple non-smooth regularizers

We consider composite optimization problems in which the objective function is given by the sum of a smooth convex term and multiple, potentially non-differentiable, convex regularizers. We show that a primal-dual method based on the proximal augmented Lagrangian, which was originally introduced for problems with two components, can be directly extended to this multi-block case. Moreover, we prove that the continuous-time primal-dual dynamics resulting from the proximal augmented Lagrangian are globally asymptotically stable even in the multi-block case if the set of equilibrium points is compact. This is in contrast to ADMM where additional assumptions, e.g., strong convexity of some components, are required. We then examine three-block problems with two non-smooth regularizers and establish global asymptotic stability of splitting dynamic resulting from the proximal augmented Lagrangian.