The non-convex problems of the downlink channel can be transformed into the ones of the dual uplink channel in TDD systems by Uplink-Downlink duality so that they are greatly simplified and numerically traced. With Lagrangian duality, the duality on beamforming and capacity region under sum power constraint and per-antenna power constraint are deduced and analyzed respectively. The software package CVX relevant to convex optimization is used to simplify the problem modeling. The analysis shows that uplink-downlink duality is in fact equivalent to Lagrangian duality. The min-max characterization is more general than uplink-downlink duality. The dual beamforming problem under per-antenna power constraint is simulated with MATLAB using iterative algorithm and interior point algorithm. Simulation results show that non-convex problems of the downlink channel can be resolved as a convex problem with Uplink-Downlink duality in the dual uplink channel.