Matrix-Monotonic Optimization-Part I: Single-Variable Optimization

Chengwen Xing, Shuai Wang, Sheng Chen, Shaodan Ma, H. Vincent Poor, Lajos Hanzo

Research output: Contribution to journalArticlepeer-review

30 Scopus citations


Matrix-monotonic optimization exploits the monotonic nature of positive semi-definite matrices to derive optimal diagonalizable structures for the matrix variables of matrix-variable 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 multiple-input 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 matrix-monotonic optimization framework can be used for deriving the optimal structures of the optimal matrix variables.

Original languageEnglish (US)
Article number9256999
Pages (from-to)738-754
Number of pages17
JournalIEEE Transactions on Signal Processing
StatePublished - 2021
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Signal Processing
  • Electrical and Electronic Engineering


  • Matrix-monotonic optimization
  • majorization theory
  • optimal structures
  • transceiver optimization


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