Matrix-monotonic optimization: Part I: single-variable optimization

Matrix-monotonic optimization exploits the monotonic nature of positive semi-definite matrices to derive optimal diagonalizable structures for the matrix variables of matrixvariable 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 our work, a comprehensive framework of the applications of matrix-monotonic optimization to multipleinput multiple-output (MIMO) transceiver design is provided for a series of specific performance metrics under various linear constraints. This framework consists of two parts, i.e., Part-I for single-variable optimization and Part-II for multi-variable optimization. In this paper, single-variable matrix-monotonic optimization is investigated under various power constraints and 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 matrixmonotonic optimization framework can be used for deriving the optimal structures of the optimal matrix variables