Analysis of distributed ADMM algorithm for consensus optimisation over lossy networks

Alternating direction method of multipliers (ADMM) is a popular convex optimisation algorithm, which is implemented in a distributed manner. Applying this algorithm to consensus optimisation problem, where a number of agents cooperatively try to solve an optimisation problem using locally available data, leads to a fully distributed algorithm which relies on local computations and communication between neighbours. In this study, the authors analyse the convergence of the distributed ADMM algorithm for solving a consensus optimisation problem over a lossy network, whose links are subject to failure. They present and analyse two different distributed ADMM-based algorithms. The algorithms are different in their network connectivity, storage and computational resource requirements. The first one converges over a sequence of networks which are not the same but remains connected over all iterations. The second algorithm is convergent over a sequence of different networks whose union is connected. The former algorithm, compared to the latter, has lower computational complexity and storage requirements. Numerical experiments confirm the proposed theoretical analysis.