Two-way relay beamforming design: Proportional fair and max-min rate fair approaches using POTDC

The challenge in designing relay beamforming in two-way relaying systems is the non-convex nature of the corresponding optimization problem. In this work, we concentrate on the mathematical issues of such design for the cases when the max-min rate and proportional fairness are used as the design criteria. We show that the corresponding optimization problems belong to the class of difference-of-convex functions (DC) programming problems. Due to the specific structure of the corresponding DC problems, they can be efficiently addressed by using the polynomial-time DC (POTDC) algorithm which guarantees to find the Karush-Kuhn-Tucker (KKT) optimal point in polynomial-time. We have also shown earlier that the question of global optimality of the POTDC algorithm boils down to a simple numerical convexity check for a certain one-dimensional optimal value function.