Lyapunov analysis of a distributed optimization scheme

We analyze the convergence of the distributed multi-agent optimization scheme originally proposed in [1]. In this scheme, a number of agents cooperate to estimate the minimum of the sum of their locally-known cost functions. We consider a special case for which the collective cost function is strongly convex and where the agent communication graph is fixed. Whereas the analysis in [1] focuses on the suboptimality of the Cesàro averages of the agents' sequences, we establish explicit ultimate bounds on the agents' estimation errors themselves. We demonstrate that the collective optimum is globally practically asymptotically stable for this algorithm.