Convergence analysis of the simplified distributed interference pricing in wireless networks

We consider the distributed pricing (DP) algorithm for adapting transmit beamforming vectors in a multi-user multiple-input multiple-output interference channel. Our objective is to maximize the sum of all user utilities. Each user announces an interference price which reflects the marginal cost of interference from other users. A particular user then updates its beamforming vector to maximize his utility minus the interference cost to other users. By local linearization of all user utilities, the beamformer update has a closed-form solution which is given by the dominant eigenvector of a matrix. We show that if the sum utility is quasiconvex with respect to the power gain vectors of the individual users, then the simplified DP algorithm must monotonically converge to a stationary point, where the power gain vector of a particular user is defined as the set of interference powers originating from that user´s transmitter. We then demonstrate for the sum rate utility, that the quasiconvexity condition is only satisfied for the two user case but not for the general setting.