This paper focuses on how to design the robust beamforming vectors of multiple peer-to-peer relay networks. In the context of imperfect channel state information at the relays, our goal is to minimize the transmit power at relay nodes subject to the quality of service at each receiver. A method based on real value Lorentz-positive maps and linear matrix inequalities is proposed, which converts the transmit power minimization problem at relay nodes into a convex optimization problem. The proposed method overcomes the disadvantages of traditional methods based on semi-definite programming relaxation, which sometimes result in solutions with rank greater than one. Simulation results show that the solution of the proposed method is consistent with those of conventional method based on semi-definite programming relaxation.

To reduce active relays in peer-to-peer relay networks, a sparsity promoting penalty term is introduced into the objective function to obtain beamforming weights through minimization of the total relay transmit power while the signal-to-interference-plus-noise ratio (SINR) at the destinations are guaranteed to be above certain thresholds. To deal with the non-convexity of the problem, we use semidefinite relaxation to turn this problem into a semidefinite programming (SDP) problem. Then we can efficiently solve the SDP problem using interior point method based software tools. Simulation results show that the proposed method can efficiently conduct relay selection with a mild increase in minimum total transmit power of the relays compared with the traditional full cooperative relay network.