Communication reduction in distributed optimization via estimation of the proximal operator

We introduce a reduced-communication distributed optimization scheme based on estimating the solution to a proximal minimization problem. Our proposed setup involves a group of agents coordinated by a central entity, altogether operating in a collaborative framework. The agents solve proximal minimization problems that are hidden from the central coordinator. The scheme enables the coordinator to construct a convex set within which the agents' optimizers reside, and to iteratively refine the set every time that an agent is queried. We analyze the quality of the constructed sets by showing their connections to the {\epsilon}-subdifferential of a convex function and characterize their size. We prove convergence results related to the solution of such distributed optimization problems and we devise a communication criterion that embeds the proposed scheme in the Alternating Direction Method of Multipliers (ADMM). The developed scheme demonstrates significant communication reduction when applied to a microgrid setting