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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 are subject to not only a sum power constraint but also interference power constraints. We transform this multiconstraint 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. Exploiting the duality result, we develop an efficient subgradient based iterative algorithm to solve the equivalent problem and show that the developed algorithm converges to a globally optimal solution. Simulation results are provided to corroborate the effectiveness of the proposed algorithm.