  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.
