Matrix-Monotonic Optimization Part II: Multi-Variable Optimization

TL;DR

This paper extends matrix-monotonic optimization to multiple matrix variables, deriving optimal structures for complex multi-variable problems in MIMO systems, sensor networks, and relaying networks, with validation through simulations.

Contribution

It demonstrates that matrix-monotonic optimization applies to multi-variable problems, providing a systematic way to derive optimal structures and simplify complex multi-variable matrix optimization tasks.

Findings

Optimal precoding matrices for MU-MIMO uplink under power constraints.

Optimal signal compression matrices in distributed sensor networks.

Transceiver designs for multi-hop MIMO relaying with imperfect CSI.

Abstract

In contrast to Part I of this treatise [1] that focuses on the optimization problems associated with single matrix variables, in this paper, we investigate the application of the matrix-monotonic optimization framework in the optimization problems associated with multiple matrix variables. It is revealed that matrix-monotonic optimization still works even for multiple matrix-variate based optimization problems, provided that certain conditions are satisfied. Using this framework, the optimal structures of the matrix variables can be derived and the associated multiple matrix-variate optimization problems can be substantially simplified. In this paper, several specific examples are given, which are essentially open problems. Firstly, we investigate multi-user multiple-input multiple-output (MU- MIMO) uplink communications under various power constraints. Using the proposed framework, the optimal structures of the precoding matrices at each user under various power constraints can be derived. Secondly, we considered the optimization of the signal compression matrices at each sensor under various power constraints in distributed sensor networks. Finally, we investigate the transceiver optimization for multi-hop amplify-and-forward (AF) MIMO relaying networks with imperfect channel state information (CSI) under various power constraints. At the end of this paper, several simulation results are given to demonstrate the accuracy of the proposed theoretical results.