  In this paper, we consider a cognitive radio (CR) network in which the
unlicensed (secondary) users are allowed to concurrently access the spectrum
allocated to the licensed (primary) users provided that their interference to
the primary users (PUs) satisfies certain constraints. We study a weighted sum
rate maximization problem for the secondary user (SU) multiple input multiple
output (MIMO) broadcast channel (BC), in which the SUs have not only the sum
power constraint but also interference constraints. We first transform this
multi-constraint maximization problem into its equivalent form, which involves
a single constraint with multiple auxiliary variables. Fixing these multiple
auxiliary variables, we propose a duality result for the equivalent problem.
Our duality result can solve the optimization problem for MIMO-BC with multiple
linear constraints, and thus can be viewed as an extension of the conventional
results, which rely crucially on a single sum power constraint. Furthermore, we
develop an efficient sub-gradient based iterative algorithm to solve the
equivalent problem and show that the developed algorithm converges to a
globally optimal solution. Simulation results are further provided to
corroborate the effectiveness of the proposed algorithm.
