  We consider the problem of achieving average consensus among multiple agents,
where the inter-agent communication network is depicted by a graph. We consider
the discrete-time consensus protocol where each agent updates its value as a
weighted average of its own value and those of its neighbours. Given a graph,
it is known that there exists a set of 'optimal weights' such that the agents
reach average consensus asymptotically with an optimal rate of convergence.
However, existing methods require the knowledge of the entire graph to compute
these optimal weights. We propose a method for each agent to compute its set of
optimal weights locally, i.e., each agent only has to know who are its
neighbours. The method is derived by solving a matrix norm minimization problem
subject to linear constraints in a distributed manner using the Alternating
Direction Method of Multipliers (ADMM). We illustrate our results using
numerical examples and compare our method with an existing method called the
Metropolis weights, which are also computed locally.
