The Successive Approximation Approach for NUM Frameworks with Elastic and Inelastic Traffic

TL;DR

This paper introduces a successive approximation method for solving nonconvex NUM problems with mixed elastic and inelastic traffic, providing a distributed algorithm that converges to a suboptimal or near-optimal solution regardless of link capacity.

Contribution

It proposes a novel distributed approximation algorithm for nonconvex NUM problems, extending to joint rate and power optimization in wireless networks with log-concave utilities.

Findings

The algorithm converges to a suboptimal solution for all link capacities.

It often converges to the global optimum in experiments.

The method applies to any log-concave utility functions.

Abstract

The concave utility in the Network Utility Maximization (NUM) problem is only suitable for elastic flows. However, the networks with the multiclass traffic, the utility of inelastic traffic is usually represented by the sigmoidal function which is a nonconcave function. Hence, the basic NUM problem becomes a nonconvex optimization problem. Solving the nonconvex NUM distributively is a difficult problem. The current works utilize the standard dual-based algorithm for the convex NUM and find the criteria for the global optimal convergence of the algorithm. It turns out that the link capacity must higher than a certain value to achieve the global optimum. We propose a new distributed algorithm that converges to the suboptimal solution of the nonconvex NUM for all of link capacity. We approximate the logarithm of the original problem to the convex problem which is solved efficiently by the standard dual-base distributed algorithm. After a sequence of approximations, the solutions converge to the KKT solution of the original problem. In many of our experiments, it also converges to the global optimal solution of the NUM. Moreover, we extend our work to solve the joint rate and power NUM problem with elastic and inelastic traffic in a wireless network. Our techniques can be applied to any log-concave utilities.