Matrix-monotonic optimization exploits the monotonic nature of positive
semi-definite matrices to derive optimal diagonalizable structures for the
matrix variables of matrix-variate optimization problems. Based on the optimal
structures derived, the associated optimization problems can be substantially
simplified and underlying physical insights can also be revealed. In this
paper, a comprehensive overview of the applications of matrix-monotonic
optimization to multiple-input multiple-output (MIMO) transceiver design is
provided under various power constraints, and matrix-monotonic optimization is
investigated for various types of channel state information (CSI) condition.
Specifically, three cases are investigated: 1)~both the transmitter and
receiver have imperfect CSI; 2)~ perfect CSI is available at the receiver but
the transmitter has no CSI; 3)~perfect CSI is available at the receiver but the
channel estimation error at the transmitter is norm-bounded. In all three
cases, the matrix-monotonic optimization framework can be used for deriving the
optimal structures of the optimal matrix variables. Furthermore, based on the
proposed framework, three specific applications are given under three types of
power constraints. The first is transceiver optimization for the multi-user
MIMO uplink, the second is signal compression in distributed sensor networks,
and the third is robust transceiver optimization of multi-hop
amplify-and-forward cooperative networks.