A popular technique of designing multiple-input multiple-output (MIMO) communication systems relies on optimizing the positive semidefinite covariance matrix at the source. In this paper, a unified MIMO optimization framework based on the Karush-Kuhn-Tucker (KKT) conditions is proposed. In this framework, with the aid of matrix optimization theory, Theorem 1 presents a generic optimal transmit covariance matrix for MIMO systems with diverse objective functions subject to various power constraints and different levels of channel state information (CSI). Specifically, Theorem 1 fundamentally reveals that for a diverse family of MIMO systems, the optimal transmit covariance matrices associated with different objective functions under various power constraints can be derived in a unified generic water-filling-like form. When applying Theorem 1 to the case of multiple general power constraints, we firstly equivalently transform multiple power constraints into a single counterpart by introducing multiple weighting factors based on Pareto optimization theory. The optimal weighting factors can be found by the proposed modified subgradient method. On the other hand, for the imperfect MIMO system with statistical CSI errors, we firstly address the non-convexity of the robust optimization problem by following the idea of alternating optimization. Finally, our numerical results verify the optimal solution structure in Theorem 1 and the global optimality of the proposed modified subgradient method, as well as demonstrate the performance advantages of the proposed alternating optimization algorithm.

中文翻译：

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基于 KKT 条件的统一 MIMO 优化框架

设计多输入多输出 (MIMO) 通信系统的一种流行技术依赖于在源处优化正半定协方差矩阵。本文提出了一种基于Karush-Kuhn-Tucker（KKT）条件的统一MIMO优化框架。在此框架下，借助矩阵优化理论，定理 1为具有不同目标函数的 MIMO 系统提供了一个通用的最优传输协方差矩阵，受各种功率约束和不同级别的信道状态信息 (CSI) 影响。具体来说，定理 1从根本上揭示了对于不同的 MIMO 系统系列，在各种功率约束下与不同目标函数相关的最佳传输协方差矩阵可以以统一的通用注水形式导出。申请时定理 1对于多个一般功率约束的情况，我们首先基于帕累托优化理论引入多个权重因子，将多个功率约束等价转化为单个对应项。可以通过所提出的改进的次梯度方法找到最佳加权因子。另一方面，对于具有统计CSI误差的不完善的MIMO系统，我们首先遵循交替优化的思想来解决鲁棒优化问题的非凸性。最后，我们的数值结果验证了最优解结构定理 1 以及所提出的改进次梯度方法的全局最优性，并证明了所提出的交替优化算法的性能优势。