  This paper considers weighted sum rate maximization constrained with a per
base station (BS) antenna power problem for multiuser multiple-input
multiple-output (MIMO) systems. For this problem, we propose new
downlink-uplink duality based solution. We solve the problem as follows. First,
by introducing additional optimization variables, we reformulate our problem
into an equivalent problem that incorporates a weighted sum mean-square-error
(MSE) term. Second, we establish novel weighted sum MSE duality. The duality is
established by modifying the input covariance matrix of the dual uplink
problem, and formulating the noise covariance matrix of the uplink channel as a
fixed point function. Third, we optimize the introduced variables and powers in
the downlink channel by a Geometric Program (GP) method. Fourth, using the
duality result and the solution of GP, we apply alternating optimization
technique to solve the original downlink problem. In our simulation results, we
have observed that the proposed duality based solution utilizes less power than
that of existing algorithms.
