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Despite significant research efforts in beamforming, the maximum achievable downlink throughput with beamforming in a multi-cell environment is not available due to difficulty in finding optimal downlink beamforming. Thus, to reformulate the problem into a more solvable form, we derive dual uplink throughput optimization problem to multi-cell downlink beam- forming throughput maximization with per-base station (BS) power constraints based on Lagrangian duality. The optimal downlink beamforming is shown to be a minimum mean squared error (MMSE) beamforming in the dual uplink. It is also shown that the dual uplink problem achieves the same optimal throughput as the primal downlink problem.