Complex-variable matrix optimization problems (CMOPs) in Frobenius norm emerge in many areas of applied mathematics and engineering applications. In this letter, we focus on solving CMOPs by iterative methods. For unconstrained CMOPs, we prove that the gradient descent (GD) method is feasible in the complex domain. Further, in view of reducing the computation complexity, constrained CMOPs are solved by a projection gradient descent (PGD) method. The theoretical analysis shows that the PGD method maintains a good convergence in the complex domain. Experiment results well support the theoretical analysis.

翻译：在应用数学和工程应用的众多领域中，出现了基于Frobenius范数的复变量矩阵优化问题（CMOPs）。本文重点研究通过迭代方法求解CMOPs。对于无约束CMOPs，我们证明了梯度下降（GD）方法在复数域中的可行性。进一步，为降低计算复杂度，采用投影梯度下降（PGD）方法求解约束CMOPs。理论分析表明，PGD方法在复数域中保持良好的收敛性。实验结果充分支持了理论分析。