  We address the problem of a multi-user relay network, where multiple
single-antenna node pairs want to exchange information by using a multiple
antenna relay node. Due to the half-duplex constraint of the relay, the
exchange of information takes place in two steps. In the first step, the nodes
transmit their data to the relay, while in the second step, the relay is
broadcasting the data by using linear and non-linear precoding strategies. We
focus on the second step in this paper. We first consider the problem of
maximizing the overall rate achievable using linear and dirty-paper type
precoding strategies at the relay. Then, we consider minimizing the total power
at the relay subject to individual SINR constraints using the same strategies
at the relay. We show that the downlink-uplink duality does not hold for the
setup considered here, which is a somewhat surprising result. We also show that
the beamforming strategy which is optimal in the single-pair case performs very
well in the multi-pair case for practically relevant SNR. The results are
illustrated by numerical simulations.
