A Proximal Atomic Coordination Algorithm for Distributed Optimization

We present a unified framework for distributed convex optimization using an algorithm called proximal atomic coordination (PAC). PAC is based on the prox-linear approach and we prove that it achieves convergence in both objective values and distance to feasibility with rate o(1/τ), where τ is the number of algorithmic iterations. We further prove that linear convergence is achieved when the objective functions are strongly convex and strongly smooth with condition number κ f , with the number of iterations on the order of square-root of κ f . This convergence rate matches the best known convergence results for distributed algorithms based on the popular alternating direction method of multipliers (ADMM) method. We also demonstrate how various decomposition strategies and coordination graphs relate to the convergence rate of PAC. We then discuss the advantages of PAC using the notions of privacy and algorithmic complexity, the latter of which involves complexities in iteration, computation, communication and storage. The theoretical results are validated using a power distribution grid in the context of optimization of power flow.