neighborhood. A comparison between [@Zhu09-DistributedInNetworkChannelCoding] and [@Schizas08-ConsensusAdHocWSNsPartI] for the average consensus problem is provided in [@Erseghe11-FastConsensusByADMM]; it is reported that both methods have similar convergence rates (and thus [@Schizas08-ConsensusAdHocWSNsPartI] uses twice as much communications as [@Zhu09-DistributedInNetworkChannelCoding]), while [@Zhu09-DistributedInNetworkChannelCoding] shows more resilience to noise. The algorithm we propose in this paper is also based on ADMM, but applied to a new reformulation of . We show through extensive simulations that the proposed algorithm requires less communications than any of the previous approaches.

All the above algorithms can solve  in a distributed way and will be briefly described in section \[Sec:RelatedAlgs\]. There are, however, other algorithms that can solve , but are not distributed. For example, [@Boyd11-ADMM] applies ADMM to distributed scenarios, but the resulting algorithms are not distributed in the sense that they require a central node or a special network topology. Our algorithm and the ones described before, in contrast, are decentralized and can run on any connected network topology.

[[**Contributions.**]{}]{} We propose a distributed algorithm for solving . This algorithm is an extension of our prior work [@Mota12-DistributedBP], where we solved a particular instance of , namely Basis Pursuit [@AtomicDecompBP]. Here, we first show that the algorithm in [@Mota12-DistributedBP] 