Minimizing Sum-MSE Implies Identical Downlink and Dual Uplink Power Allocations

In the multiuser downlink, power allocation for linear precoders that minimize the sum of mean squared errors under a sum power constraint is a non-convex problem. Many existing algorithms solve an equivalent convex problem in the virtual uplink and apply a transformation based on uplink-downlink duality to find a downlink solution. In this letter, we analyze the optimality criteria for the power allocation subproblem in the virtual uplink, and demonstrate that the optimal solution leads to identical power allocations in the downlink and virtual uplink. We thus extend the known duality results and, importantly, simplify the existing algorithms used for iterative transceiver design.