On the geometric convergence rate of distributed economic   dispatch/demand response in power networks
  Motivated by potential applications in power systems, we study a problem of optimizing a sum of $n$ convex functions on dynamic networks of $n$ nodes when each function is known to only a single node. The nodes' variables, while satisfy their local constraints, are coupled through a linear constraint. Our main contribution is to design a fully distributed primal-dual method for this problem. Under some fairly standard assumptions on objective functions, strong convexity and smoothness, we provide an explicit analysis for the convergence rate of our method on different networks. In particular, the nodes variables achieve a geometric convergence to the optimal with the associated convergence time scales quartically in the number of nodes on any sequence of time-varying undirected graphs satisfying a long-term connectivity condition. Moreover, this convergence time is constant independent on the number of nodes when the network is a b-regular simple graph with $b\geq 3$. Finally, to show the effectiveness of our method we also simulate a number of studies on economic dispatch problems and demand response problems in power systems. 
Thinh T. Doan, Alex Olshevsky
math.OC