Matrix-variate optimization plays a central role in advanced wireless system designs. In this paper, we aim to explore optimal solutions of matrix variables under two special structure constraints using complex matrix derivatives, including diagonal structure constraints and constant modulus constraints, both of which are closely related to the state-of-the-art wireless applications. Specifically, for diagonal structure constraints mostly considered in the uplink multi-user single-input multiple-output (MU-SIMO) system and the amplitude-adjustable intelligent reflecting surface (IRS)-aided multiple-input multiple-output (MIMO) system, the capacity maximization problem, the mean-squared error (MSE) minimization problem and their variants are rigorously investigated. By leveraging complex matrix derivatives, the optimal solutions of these problems are directly obtained in closed forms. Nevertheless, for constant modulus constraints with the intrinsic nature of element-wise decomposability, which are often seen in the hybrid analog-digital MIMO system and the fully-passive IRS-aided MIMO system, we firstly explore inherent structures of the element-wise phase derivatives associated with different optimization problems. Then, we propose a novel alternating optimization (AO) algorithm with the aid of several arbitrary feasible solutions, which avoids the complicated matrix inversion and matrix factorization involved in conventional element-wise iterative algorithms. Numerical simulations reveal that the proposed algorithm can dramatically reduce the computational complexity without loss of system performance.

翻译：矩阵变量优化在现代无线系统设计中占据核心地位。本文旨在利用复数矩阵导数探索矩阵变量在两种特殊结构约束下的最优解，包括对角结构约束和恒模约束，这两种约束均与前沿无线应用密切相关。具体而言，针对主要考虑于上行多用户单输入多输出（MU-SIMO）系统及振幅可调智能反射面（IRS）辅助多输入多输出（MIMO）系统中的对角结构约束，本文严格研究了容量最大化问题、均方误差（MSE）最小化问题及其变体。通过利用复数矩阵导数，这些问题的最优解可直接以闭式形式获得。然而，对于具有内在元素可分解特性的恒模约束（常见于混合模拟-数字MIMO系统与全无源IRS辅助MIMO系统），我们首先探索了不同优化问题中与元素级相位导数相关的内在结构。随后，提出了一种借助多个任意可行解的新型交替优化（AO）算法，该算法避免了传统逐元素迭代算法中复杂的矩阵求逆与矩阵分解。数值仿真表明，所提算法能在不损失系统性能的前提下显著降低计算复杂度。