Maximizing sum rate and minimizing MSE on multiuser downlink: Optimality, fast algorithms and equivalence via max-min SINR

Maximizing the minimum weighted signal-to-interference-and-noise ratio (SINR), minimizing the weighted sum mean-square error (MSE) and maximizing the weighted sum rate in a multiuser downlink system are three important performance objectives in nonconvex joint transceiver and power optimization, where all the users have a total power constraint. We show that, through connections with the nonlinear Perron-Frobenius theory, jointly optimizing power and beamformers in the max-min weighted SINR problem can be solved optimally in a distributed fashion. Then, connecting these three performance objectives through the arithmetic-geometric mean inequality and nonnegative matrix theory, we solve the weighted sum MSE minimization and the weighted sum rate maximization in the weak interference regimes using fast algorithms. In the general case, we first establish optimality conditions to the weighted sum MSE minimization and the weighted sum rate maximization problems and provide their further connection to the max-min weighted SINR problem. We then propose a distributed weighted proportional SINR algorithm that leverages our fast max-min weighted SINR algorithm to solve for local optimal solution of the two nonconvex problems, and give conditions under which global optimality is achieved. Numerical results are provided to complement the analysis.