I have an optimization problem as following \begin{equation} \begin{aligned} \min\limits_{H,H^*}||y-AHx-BH^*x^*||^2\\ \text{such that:}\qquad Ax+Bx^*=g \end{aligned} \end{equation}
in which
- $(.)^*$ means conjugation.
- $A$ and $B$ are $N \times N$ diagonal matrices.
- $y$, $g$, and $x$ are $N \times 1$ column vectors
- $H$ is an upper triangular matrix
- Both vectors and matrices are generally complex
In the optimization problem, only $y$ and $g$ are known because they can be observed from experiments, while $A$, $B$, $x$ are unknown but having the constraint as in $g$
I've already tried to expand the $||.||$ in the objective function and then partially derived this function with $h_1,h_2,\text{etc}$, which are the elements of $H$, but it was too complicated and I got stuck.
Have any idea to resolve this problem? Thank you so much