  Recently, researchers showed that dirty paper coding (DPC) is the optimal
transmission strategy for multiple-input multiple-output broadcast channels
(MIMO-BC). In this paper, we study how to determine the maximum weighted sum of
DPC rates through solving the maximum weighted sum rate problem of the dual
MIMO multiple access channel (MIMO-MAC) with a sum power constraint. We first
simplify the maximum weighted sum rate problem such that enumerating all
possible decoding orders in the dual MIMO-MAC is unnecessary. We then design an
efficient algorithm based on conjugate gradient projection (CGP) to solve the
maximum weighted sum rate problem. Our proposed CGP method utilizes the
powerful concept of Hessian conjugacy. We also develop a rigorous algorithm to
solve the projection problem. We show that CGP enjoys provable convergence,
nice scalability, and great efficiency for large MIMO-BC systems.
