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：复杂的变量矩阵优化问题（CMOPs）在应用数学和工程应用领域中经常出现。在这封信中，我们专注于通过迭代方法解决CMOPs。对于无约束CMOPs，我们证明了梯度下降（GD）方法在复杂域中是可行的。此外，为了降低计算复杂度，受限制的CMOPs通过投影梯度下降（PGD）方法来解决。理论分析表明，PGD方法在复杂域中保持良好的收敛性。实验结果很好地支持了理论分析。