  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.
