Fast primal-dual projected linear iterations for distributed consensus in constrained convex optimization

In this paper we study the constrained consensus problem, i.e. the problem of reaching a common point from the estimates generated by multiple agents that are constrained to lie in different constraint sets. First, we provide a novel formulation of this problem as a convex optimization problem but with coupling constraints. Then, we propose a primal-dual decomposition method for solving this type of coupled convex optimization problems in a distributed fashion given restrictions on the communication topology. The proposed algorithm is based on consensus principles (as an efficient strategy for information fusion in networks) in combination with local subgradient updates for the primal-dual variables. We show, for the first time, that the nonnegative weights corresponding to the consensus process can be interpreted as dual variables and thus they can be updated using arguments from duality theory. Therefore, in our algorithm the weights are updated following some precise rules, while in most of the existing distributed algorithms based on consensus principles the weights have to be tuned. Preliminary simulation results show that our algorithm works, an average, ten times faster than some existing methods.