Complexity of gradient projection method for optimal routing in data networks

In this paper, we derive a time-complexity bound for the gradient projection method for optimal routing in data networks. This result shows that the gradient projection algorithm of the Goldstein-Levitin-Poljak type formulated by Bertsekas converges to within epsilon in relative accuracy in O(epsilon(-2)h(min)N(max)) number of iterations, where N-max is the number of paths sharing the maximally shared link, and h(min) is the diameter of the network. Based on this complexity result, we also show that the one-source-at-a-time update policy has a complexity bound which is O(n) times smaller than that of the all-at-a-time update policy, where n is the number of nodes in the network. The result of this paper argues for constructing networks with low diameter for the purpose of reducing complexity of the network control algorithms. The result also implies that parallelizing the optimal routing algorithm over the network nodes is beneficial.