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In this paper, we consider optimization of a two-way multiple-input multiple-output (MIMO) amplify-and-forward relay network which consists of a pair of transceivers and several relay nodes. Multiple antennas are equipped on the transceivers and relays. Multiple access broadcast scheme which finishes communication in two time slots is considered. In the first time slot, signals received by the relays are scaled by several beamforming matrices. In the second time slot, the relays transmit the scaled signals to the two transceivers. Upon receiving these signals, a MIMO equalizer is implemented at each transceiver to recover the desired signal. In this paper, zero forcing equalizers are used. Joint optimization of the beamforming matrices and the equalizers are realized using the following criteria: 1) the total relay transmission power is minimized subject to the minimal output signal-to-noise ratio (SNR) constraint at each transceiver, 2) the minimal output SNR of the two transceivers is maximized subject to total relay transmission power constraint, and 3) the minimal output SNR of the two transceivers is maximized subject to individual relay transmission power constraint. It is shown that the proposed optimization problems can be formulated as the second-order cone programming problems which can be solved efficiently. The validity of the proposed algorithm is verified by computer simulations.

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