A Distributed Proximal Primal-Dual Algorithm for Nonsmooth Optimization with Coupling Constraints

This paper develops a novel algorithm that tackles a general distributed convex optimization problem with a nonsmooth objective function and a series of coupling nonlinear inequality and linear equality constraints. To address such a problem, we first design a distributed proximal algorithm for problems with coupling equality constraints only, and then extend this algorithm to problems with both inequality and equality constraints by incorporating a virtual-queue-based method. The resulting algorithm, referred to as Proximal Primal-Dual Algorithm (PPDA), is shown to achieve O(1/k) rates of convergence with respect to both optimality and feasibility. Compared to the alternative methods in the literature, PPDA eliminates their assumptions on the smoothness of the objective function and achieves a stronger convergence rate result. In addition, PPDA exhibits faster convergence in a couple of numerical examples.