A Proximal Atomic Coordination Algorithm for Distributed Optimization

In this article, 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/
$\tau$
), where
$\tau$
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
$\kappa _{f}$
, with the number of iterations on the order of square-root of
$\kappa _{f}$
. We demonstrate how various decomposition strategies and coordination graphs relate to the convergence rate of PAC. We then compare this convergence rate with that of a distributed algorithm based on the popular alternating direction method of multipliers (ADMMs) method. We further compare the algorithmic complexities of PAC to ADMM and enumerate the ensuing advantages. Finally, we demonstrate yet another advantage of PAC related to privacy. All theoretical results are validated using a power distribution grid model in the context of the optimal power flow problem.