为了求在广义约束$GX=H, \ WY=U$条件下矩阵方程$AXB+CYD=E$的最佳逼近解, 提出了一种迭代算法。该算法思路是首先分别求出目标函数$F(X, Y)={{\left\| E-AXB-CYD \right\|}^{2}}$在矩阵$X, \ Y$处的梯度; 然后将负梯度分别投影到凸约束集中得到${g}_{X}$和${g}_{Y}$; 最后按照共轭梯度法思想, 基于${{g}_{X}}$和${{g}_{Y}}$在可行域上再构建搜索方向${{d}_{X}}$和${{d}_{Y}}$。理论表明对于任给一个满足广义约束的一类特殊初始矩阵对$({{X}^{(1)}}, {{Y}^{(1)}})$, 算法能够在有限迭代步内得到约束条件下矩阵方程$AXB+CYD=E$的极小范数最小二乘解。另外通过求矩阵方程$A\tilde{X}B+C\tilde{Y}D=\tilde{E}$的极小范数最小二乘解可得给定逼近矩阵对$\left( \bar{X}, \, \bar{Y} \right)$的最佳逼近解, 其中$\tilde{E}=E-A\bar{X}B-C\bar{Y}D$。数值例子表明该算法不仅可以解决广义约束条件下矩阵方程的最佳逼近解, 也可以解决特殊约束条件下方程的最佳逼近解。

In this paper, we present an iterative algorithm to calculate the optimal approximation solution pair of the matrix equations $AXB+CYD=E $with constraint conditions $GX=H $and $\ WY=U$. The idea of the algorithm is to first find the gradient of the objective function $F(X, Y)={{\left\| E-AXB-CYD \right\|}^{2}}$ at the matrix $X $and $Y$, and then project the negative gradient to the convex constraint set respectively to obtain ${{g}_{X}}$ and ${{g}_{Y}}$. Finally, according to the idea of conjugate gradient method, the search directions ${{d}_{X}}$ and ${{d}_{Y}}$ are reconstructed on the feasible domain based on ${{g}_{X}}$ and ${{g}_{Y}}$. The theory shows that the algorithm can obtain the minimal norm least squares solution pair of the matrix equation $AXB+CYD=E $under the constraint conditions in finite iterative steps for any special class of initial matrix pair $({{X}^{(1)}}, {{Y}^{(1)}}) $satisfying the constraint conditions. In addition, the optimal approximation solution pair to a given matrix pair $\left( \bar{X}, \, \bar{Y} \right) $can be obtained by finding the minimal norm least squares solution pair of a new matrix equations $A\tilde{X}B+C\tilde{Y}D=\tilde{E}$, where $\tilde{E}=E-A\bar{X}B-C\bar{Y}D$. Numerical examples show that the algorithm can not only solve the optimal approximation solutions of matrix equations under generalized constraints, but also solve the optimal approximation solutions of equations under special constraints.