Direct derivation of the gradient method for network utility maximization in broadcast channels and its application

In broadcast channels, network utility maximization formulates a scheduling problem in order to maximize a sum of given utility functions. It is known that the network utility maximization is achieved if a gradient method is used. In this paper, however, we show that the network utility maximization is achieved only if a gradient method is used. That proves the equivalence between a problem formulated by the network utility maximization and a problem with a gradient method. The gradient method simplifies the object function of a scheduling problem by modifying utility functions to a gradient form, so that it makes easy to deal with the problem. We apply the gradient method to a problem with utility functions given by generalized proportional fairness. It is revealed that using the gradient method for the generalized proportional fairness is equivalent to applying a binomial approximation. Simulation results are presented with various scheduling parameters.