We consider distributed convex optimization problems that involve a separableobjective function and nontrivial functional constraints, such as Linear MatrixInequalities (LMIs). We propose a decentralized and computationally inexpensivealgorithm which is based on the concept of approximate projections. Ouralgorithm is one of the consensus based methods in that, at every iteration,each agent performs a consensus update of its decision variables followed by anoptimization step of its local objective function and local constraints. Unlikeother methods, the last step of our method is not an Euclidean projection ontothe feasible set, but instead a subgradient step in the direction thatminimizes the local constraint violation. We propose two different averagingschemes to mitigate the disagreements among the agents' local estimates. Weshow that the algorithms converge almost surely, i.e., every agent agrees onthe same optimal solution, under the assumption that the objective functionsand constraint functions are nondifferentiable and their subgradients arebounded. We provide simulation results on a decentralized optimal gossipaveraging problem, which involves SDP constraints, to complement ourtheoretical results.
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