An $O(1/k)$ Gradient Method for Network Resource Allocation Problems

We present a fast distributed gradient method for a convex optimization problem with linear inequalities, with a particular focus on the network utility maximization (NUM) problem. Most existing works in the literature use (sub)gradient methods for solving the dual of this problem which can be implemented in a distributed manner. However, these (sub)gradient methods suffer from an O(1/√k) rate of convergence (where k is the number of iterations). In this paper, we assume that the utility functions are strongly concave, an assumption satisfied by most standard utility functions considered in the literature. We develop a completely distributed fast gradient method for solving the dual of the NUM problem. We show that the generated primal sequences converge to the unique optimal solution of the NUM problem at rate O(1/k).