Optimal polynomial filtering for accelerating distributed consensus

In the past few years, the problem of distributed consensus has received a lot of attention, particularly in the framework of ad hoc sensor networks. Most methods proposed in the literature attack this problem by distributed linear iterative algorithms, with asymptotic convergence of the consensus solution. It is known that the rate of convergence depends on the second largest eigenvalue of the weight matrix. In this paper, we propose the use of polynomial filtering in order to accelerate the convergence rate. The main idea of the proposed methodology is to apply a polynomial filter that will shape the spectrum of the weight matrix by minimizing its second largest eigenvalue and therefore increase the convergence rate. We formulate the computation of the optimal polynomial as a semi-definite program (SDP) that can be efficiently and globally solved. We provide simulation results that demonstrate the validity and effectiveness of the proposed scheme in both fixed and dynamic network topologies.