I am trying to solve the following optimization problem with Lagrangian multipliers:
$\min_{x} ||Ap+Bx||^{2}\\ s.t. ||x||^{2} \leq E$
Where A is a n by m matrix with complex numbers, p is a m complex column vector, B is a n by s matrix with complex numbers and finally x is a s complex column vector. E is a scalar constraint value (like 0.5).
Then we write a Lagrangian cost function:
$\min_{x}L(x,\lambda)=||Ap+Bx||^{2}+\lambda(||x||^{2}-E)$
The solution can be found then by taking a derivative with respect to Re and Im parts of x separately, like:
$\frac{\partial{\mathbf{L}}}{\partial x}=\frac{\partial{\mathbf{L}}}{\partial x_{Re}}+j\frac{\partial{\mathbf{L}}}{\partial x_{Im}}$
Questions:
1) When we open squared norm as, for example, $||x||^{2}=x^{H}x = (x_{Re}+jx_{Im})^{H}(x_{Re}+jx_{Im})$ does it equal to :$(x_{Re}^{T}-jx_{Im}^{T})(x_{Re}+jx_{Im})$, (where T denotes to transpose) and is it mathematically correct?
2) At some point we get expressions: $x_{Re}^{T}B^{H}Ap \\ x_{Re}^{T}B^{H}Ax_{Re} \\ p^{H}A^{H}Bx_{Re} $
What are the derivatives of them in respect to $x_{Re}$ ($\frac{\partial{\mathbf{}}}{\partial x_{Re}}$) ?
On Wikipedia there is Matrix_calculus page with formulas of different derivatives, but it is mentioned that vectors and matrices are real and in our case we have complex vector and matrices. Do that rules can be used also for complex vectors/matrices?
Is there any comprehensive literature with derivatives of complex matrices/vectors?
3) As written at some papers about Lagrangian multipliers and especially in Khan Academy, L depends on two variables $x$ and $\lambda$, so solving a system of two equations, which have been found by taking two derivatives: $\frac{\partial{\mathbf{L}}}{\partial x}$ and $\frac{\partial{\mathbf{L}}}{\partial \lambda}$ it is possible to find $x$ and $\lambda$.
Will it work also in our case? Or there is any limitation of this, when dealing with complex vectors/matrices?
Is it necessary also to separate Lagrangian multiplier to real and imaginary part?, although in our case, as I understood, $\lambda$ is real and positive value and we should take only one derivative $\frac{\partial{\mathbf{L}}}{\partial \lambda}$ is not it?
Well finally $\frac{\partial{\mathbf{L}}}{\partial \lambda}$ equals to $||x||^{2}-E=0$, which is logically for me, but then I am expecting from first equation (which is the result of $\frac{\partial{\mathbf{L}}}{\partial x}$) to get x and not $||x||^{2}$ so I see the trap in this system equation and how to solve it, when in first equation there is a $x$ and in the second $||x||^{2}$.
I hope I was clear, but, please if it not so at some point, ask me, because after almost a week of trying to find the solution I am stuck and really rely on your help.
Maybe even there is a mistake of the way how I want to solve it and I am wrong at the beginning.