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We consider the communication over a set of parallel multiple-input single-output (MISO) broadcast channels with separate linear precoding on each of them. In this setup, we study the problem of fulfilling per-user quality of service constraints, expressed in terms of rates, using minimal transmit power. By means of a dual decomposition approach and a branch-and-bound algorithm solving the arising nonconvex subproblems, we find the so far unknown globally optimal solution for the case of linear precoding with time sharing. Although prohibitively complex for implementation in a practical system, the new power minimization method is highly interesting from a theoretical point of view as it can be used to quantify the performance gap between the outcome of heuristic algorithms and the theoretical limitations of the system with linear transceivers. We also extend the approach such that zero-forcing constraints can be handled.