  We consider multi-antenna base stations using orthogonal frequency division
multiple access and space division multiple access techniques to serve
single-antenna users. Some users, called real-time users, have minimum rate
requirements and must be served in the current time slot while others, called
non real-time users, do not have strict timing constraints and are served on a
best-effort basis. The resource allocation problem is to find the assignment of
users to subcarriers and the transmit beamforming vectors that maximize the
total user rates subject to power and minimum rate constraints. In general,
this is a nonlinear and non-convex program and the zero-forcing technique used
here makes it integer as well, exact optimal solutions cannot be computed in
reasonable time for realistic cases. For this reason, we present a technique to
compute both upper and lower bounds and show that these are quite close for
some realistic cases.
  First, we formulate the dual problem whose optimum provides an upper bound to
all feasible solutions. We then use a simple method to get a primal-feasible
point starting from the dual optimal solution, which is a lower bound on the
primal optimal solution. Numerical results for several cases show that the two
bounds are close so that the dual method can be used to benchmark any heuristic
used to solve this problem. As an example, we provide numerical results showing
the performance gap of the well-known weight adjustment method and show that
there is considerable room for improvement.
