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.