  We consider the problem of rate allocation in a Gaussian multiple-access
channel, with the goal of maximizing a utility function over transmission
rates. In contrast to the literature which focuses on linear utility functions,
we study general concave utility functions. We present a gradient projection
algorithm for this problem. Since the constraint set of the problem is
described by exponentially many constraints, methods that use exact projections
are computationally intractable. Therefore, we develop a new method that uses
approximate projections. We use the polymatroid structure of the capacity
region to show that the approximate projection can be implemented by a
recursive algorithm in time polynomial in the number of users. We further
propose another algorithm for implementing the approximate projections using
rate-splitting and show improved bounds on its convergence time.
