We propose a distributed, decentralized algorithm for solving separable optimization problems over a connected network of compute nodes. In a separable problem, each node has its own private function and its own private constraint set. Private means that no other node has access to it. The goal is to minimize the sum of all nodes’ private functions, constraining the solution to be in the intersection of all the private sets. Our algorithm is based on the alternating direction method of multipliers (ADMM) and requires a coloring of the network to be available beforehand. We perform numerical experiments of the algorithm, applying it to compressed sensing problems. These show that the proposed algorithm requires in general less iterations, and hence less communication between nodes, than previous algorithms to achieve a given accuracy.