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In this paper, we consider a cognitive radio (CR) network in which the unlicensed (secondary) users (SUs) 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 establish a duality result for the equivalent problem. Our duality result can be viewed as an extension of the previously known results, which depend on either a sum power constraint or per-antenna power constraints. 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. Computer simulations are also provided to corroborate the effectiveness of the proposed algorithm.