Beamforming design for multi-pair two-way relaying systems via monotonic program

This paper investigates the optimal relay beamforming matrix design problem in amplify-and-forward (AF) based multiuser two-way relay channels (TWRC). A unified framework is proposed for beamforming designs for various design criteria. The core of our framework is the solution to the classic max-min signal-to-interference-plus-noise ratio (SINR) problem for multiuser TWRCs. Treated as a max-min fractional program, the max-min SINR problem can be solved by a Dinkelbach-type algorithm with near optimal performance and fast convergence speed. Using the max-min SINR solution as a corner stone, we show that the optimal beamforming designs for maximization of weighted sum-rate, minimization of weighted sum mean-square-errors, and other many criteria can be reformulated into a monotonic program. Relying on solving a series of max-min SINR problems, a polyblock outer approximation algorithm is then developed to find the desired solutions with guaranteed convergence and near-optimal performance.