Given one optimization objective： $$ \min\limits_{U}||U^TX-Y||_F^2+\lambda tr(U^TQU)\quad s.t. U^T\Sigma U=aI $$ Where $X,Y,\Sigma$ are all known matrixs, $\lambda,a$ are constants, $Q$ is a symmetric matrix, $I$ is an identity matrix, introducing lagrange multiplier matrix $\Lambda$(a diagonal matrix), we can get Lagrangian function $L$: $$ L=tr((U^TX-Y)^T(U^TX-Y))+\lambda tr(U^TQU)+tr(\Lambda^T(aI-U^T\Sigma U)) $$ It's derivative is $\frac{\partial L}{\partial U}=(XX^T+XX^T)U-XY^T-XY^T+\lambda(QU+Q^TU)-2\Sigma U\Lambda=2XX^TU-2XY^T+2\lambda QU-2\Sigma U\Lambda$
make $\frac{\partial L}{\partial U}=0$, so $(XX^T+\lambda Q)U=XY^T+\Sigma U\Lambda$
But how to calculate $U$? and is it right? Please help me, thank you very much!
Exact Problem: Does there exist a closed-form solution for $U$ satisfying $(XX^T+\lambda Q)U=XY^T+\Sigma U\Lambda$?