  In this paper, we consider the resource allocation problem in a network with
a large number of connections which are used by a huge number of users. The
resource allocation problem under discussion is a maximization problem with
linear inequality constraints. To solve this problem we construct the dual
problem and propose to use the following numerical optimization methods for the
dual: a fast gradient method, a stochastic projected subgradient method, an
ellipsoid method, and a random gradient extrapolation method. A special focus
is made on the primal-dual analysis of these methods. For each method we
estimate the convergence rate. We also provide some modifications of these
methods in the setup of distributed computations, taking into account their
application to networks.
