Power minimization in MIMO cognitive networks using beamforming games

We consider a multi-channel multi-user cognitive radio MIMO network in which each node controls its antenna radiation directions and allocates power for each data stream by adjusting its precoding matrices. Under a noncooperative game, we optimize the set of precoding matrices (one per channel) at each node so as to minimize the total transmit power in the network. Using recession analysis and the theory of variational inequalities, we obtain sufficient conditions that guarantee the existence and uniqueness of the game's Nash Equilibrium (NE). Low-complexity distributed algorithms are also developed by exploiting the strong duality of the convex per-user optimization problem. To improve the efficiency of the NE, we introduce pricing policies that employ a novel network interference function. Existence and uniqueness of the new NE under pricing are studied. Simulations confirm the effectiveness of our joint optimization approach.