Distributed constrained convex optimization and consensus via dual decomposition and proximal minimization

We consider a general class of convex optimization problems over time-varying, multi-agent networks, that naturally arise in many application domains like energy systems and wireless networks. In particular, we focus on programs with separable objective functions, local (possibly different) constraint sets and a coupling inequality constraint expressed as the non-negativity of the sum of convex functions, each corresponding to one agent. We propose a novel distributed algorithm to deal with such problems based on a combination of dual decomposition and proximal minimization. Our approach is based on an iterative scheme that enables agents to reach consensus with respect to the dual variables, while preserving information privacy. Specifically, agents are not required to disclose information about their local objective and constraint functions, nor to assume knowledge of the coupling constraint. Our analysis can be thought of as a generalization of dual gradient/subgradient algorithms to a distributed set-up. We show convergence of the proposed algorithm to some optimal dual solution of the centralized problem counterpart, while the primal iterates generated by the algorithm converge to the set of optimal primal solutions. A numerical example demonstrating the efficacy of the proposed algorithm is also provided.