Abstract:

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.

Key words: matrix equations, optimal approximation solution, iterative algorithm, gradient projection, orthogonal vectors