Hypothesis:
- we are given 4 complex matrices denoted with $H_1, H_2, G_1$ and $G_2$.
- the 4 matrices are not necessarily square so their size is $N$ by $M$.
- we denote with $w_1$ and $w_2$ two variable complex column vectors ($M$ by $1$) with the energies constrained to some given values $E_1=w_1^Hw_1=|w_1|^2$ and $E_2=w_2^Hw_2=|w_2|^2$
What I want:
- maximize with respect to $w_1$ and $w_2$ the following real expression (find $w_1$ and $w_2$):
$$f(w_1,w_2) = (1+|H_1w_1|^2-\frac{|(G_2w_2)^H(H_1w_1)|^2}{1+|G_2w_2|^2})(1+|H_2w_2|^2-\frac{|(G_1w_1)^H(H_2w_2)|^2}{1+|G_1w_1|^2})$$
where I used the notation $()^H$ for the hermitian operator (transposition and complex conjugation) and the notation $||^2$ for the energy of a scalar or a vector ($|v|^2=v^Hv$ for a column vector or scalar).
$\quad$I tried to exploit the simmetries to reduce the expression to a Rayleigh quotient so the norm constraints do not matter. This problem emerged in the context of maximizing the sum capacity of two cell interference channel. The problem is that $w_1$ and $w_2$ appear in both subexpressions so I cannot maximize separately the two brackets. Can someone help me? Or even suggest some things to try...or if there is an analitical solution. Any help is much appreciated!
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