Blended Dynamics Approach to Distributed Optimization: Sum Convexity and Convergence Rate

Abstract: This paper studies the application of the blended dynamics approach towards distributed optimization problem where the global cost function is given by a sum of local cost functions. The benefits include (i) individual cost function need not be convex as long as the global cost function is strongly convex and (ii) the convergence rate of the distributed algorithm is arbitrarily close to the convergence rate of the centralized one. Two particular continuous-time algorithms are presented using the proportional-integral-type couplings. One has benefit of `initialization-free,' so that agents can join or leave the network during the operation. The other one has the minimal amount of communication information. After presenting a general theorem that can be used for designing distributed algorithms, we particularly present a distributed heavy-ball method and discuss its strength over other methods.