This letter considers joint beamformer assignment and power allocation (BAPA) for codebook-based downlink beamforming. The BAPA problem represents a mixed integer combinatorial program since beamformer assignments involve binary decisions and beamformer assignments of multiple users are coupled in the downlink SINR constraints. We develop a mixed integer linear program formulation of the BAPA problem, which can be solved using, e.g., the branch-and-cut method. To derive low-complexity solutions, we introduce a virtual uplink problem, in which beamformer assignments of different users are decoupled. We establish the uplink-downlink duality of the two problems and develop a customized iterative algorithm to solve the BAPA problem. Analytic studies show that the customized algorithm yields either optimal (within the desired numerical accuracy) solutions of the BAPA problem (when it is feasible), or infeasibility certificates (when infeasible). The performance of the algorithm is demonstrated with numerical examples.