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We consider the power minimization problem with per-user minimum rate constraints for parallel multiple-input multiple-output (MIMO) broadcast channels employing zero-forcing beamforming. Recent results have shown that spreading data streams across several carriers-so called carrier-cooperative (CC) transmission-can lead to a reduction of the sum transmit power in such a scenario. However, using state-of-the-art power minimization algorithms based on zero-forcing, only carrier-noncooperative (CN) solutions can be obtained. In this paper, we derive a novel algorithm that is capable of finding CC transmit strategies and can achieve a significant decrease in sum transmit power compared to a conventional zero-forcing power minimization method. The key point of the algorithm is that it combines greedy allocation of data streams, which is a popular technique to optimize zero-forcing strategies, with a gradient-based update of the filter vectors, which is a way to ensure that CC solutions can be obtained. Numerical simulations show that the advantage of the new algorithm is most pronounced in an environment where users have spectrally similar channels.