This paper address the problem of voltage regulation in power distribution networks with deep penetration of distributed energy resources (DERs) without any explicit communication between the buses in the network. We cast the problem as an optimization problem with the objective of minimizing the distance between the bus voltage magnitudes and some reference voltage profile. We present an iterative algorithm where each bus updates the reactive power injection provided by their DER. The update at a bus only depends on the voltage magnitude at that bus, and for this reason, we call the algorithm a local control algorithm. We provide sufficient conditions that guarantee the convergence of the algorithm and these conditions can be checked a priori for a set of feasible power injections. We also provide necessary conditions establishing that longer and more heavily loaded networks are inherently more difficult to control. We illustrate the operation of the algorithm through case studies involving 8-,34- and 123-bus test distribution systems.