Consensus acceleration in multiagent systems with the Chebyshev semi-iterative method

We consider the fundamental problem of reaching consensus in multiagent systems. To date, the consensus problem has been typically solved with decentralized algorithms based on graph Laplacians. However, the convergence of these algorithms is often too slow for many important multiagent applications, and thus they are increasingly being combined with acceleration methods. Unfortunately, state-of-the-art acceleration techniques require parameters that can be optimally selected only if complete information about the network topology is available, which is rarely the case in practice. We address this limitation by deriving two novel acceleration methods that can deliver good performance even if little information about the network is available. The first is based on the Chebyshev semi-iterative method and maximizes the worst-case convergence speed given that only rough bounds on the extremal eigenvalues of the network matrix are available. It can be applied to systems where agents use unreliable communication links, and its computational complexity is similar to those of simple Laplacian-based methods. This algorithm requires synchronization among agents, so we also propose an asynchronous version that approximates the output of the synchronous algorithm. Mathematical analysis and numerical simulations show that the convergence speed of the proposed acceleration methods decrease gracefully in scenarios where the sole use of Laplacian-based methods is known to be impractical.