Total power minimization for two-way networks with multi-antenna relays

We consider a two-way relay network consisting of two single-antenna transceivers and multiple multi-antenna relays. Assuming a multiple access broadcast (MABC) relaying scheme, we aim to jointly obtain the optimal relay beamforming matrices as well as the optimal transceiver transmit powers which minimize the total transmit power under given signal-to-noise-ratio (SNR) constraints at the transceivers. We show that the relay beamforming matrices have special structures which can be exploited to reduce the computational complexity of the optimization problem. We use this structure to re-cast the total power minimization problem as an unconstrained optimization problem with a dimensionality which is smaller than that of the original problem, when number of antennas in the relays is larger than 2. We then show this unconstrained problem can be solved using a two-dimensional search over the feasible set of the transceivers´ transmit powers. Indeed, the feasible set is quantized into a sufficiently fine grid. At each vertex of this grid, a quadratically constrained quadratic problem (QCQP) is solved to find the optimal values of the relay beamforming matrices corresponding to that vertex. This QCQP can be rewritten as a semi-definite relaxation (SDR) method which is guaranteed to have a rank-one solution. Once the beamforming matrices are obtained for all vertices of the grid, the vertex which results in the lowest value for the total transmit power, yields the optimal values of the transceivers´ transmit powers and the optimal relay beamforming matrices.