  We consider distributed convex optimization problems that involve a separable
objective function and nontrivial functional constraints, such as Linear Matrix
Inequalities (LMIs). We propose a decentralized and computationally inexpensive
algorithm which is based on the concept of approximate projections. Our
algorithm is one of the consensus based methods in that, at every iteration,
each agent performs a consensus update of its decision variables followed by an
optimization step of its local objective function and local constraints. Unlike
other methods, the last step of our method is not an Euclidean projection onto
the feasible set, but instead a subgradient step in the direction that
minimizes the local constraint violation. We propose two different averaging
schemes to mitigate the disagreements among the agents' local estimates. We
show that the algorithms converge almost surely, i.e., every agent agrees on
the same optimal solution, under the assumption that the objective functions
and constraint functions are nondifferentiable and their subgradients are
bounded. We provide simulation results on a decentralized optimal gossip
averaging problem, which involves SDP constraints, to complement our
theoretical results.
