Simultaneous maximization of two Generalized Rayleigh Ritz Ratios Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios $\frac{x^{H}A_1x}{x^{H}B_1x}$ and $\frac{x^{H}A_2x}{x^{H}B_2x}$. To state it formally, the problem is 
\begin{align}
\max_{x} \min\left(\frac{x^{H}A_1x}{x^{H}B_1x},\frac{x^{H}A_2x}{x^{H}B_2x}\right)
\end{align}
I would be very much happy if we can find a closed form solution for it. I would also appreciate if some one can suggest a iterative algorithm to solve it. I am not interested in converting it to a convex optimization problem even if it is possible. (But, if convertible to a convex is fruitful, then I would like to see it). 
EDIT------ 
My attempt on the problem 
Introduce a extra variable $t$ and rewrite the problem as 
\begin{align}
\max_{x,t}~t ,  
\end{align}
\begin{align}sucht~that~
\frac{x^{H}A_1x}{x^{H}B_1x} \geq t,
\frac{x^{H}A_2x}{x^{H}B_2x} \geq t
\end{align}
Since $B_1$ and $B_2$ are positive definite, one can re-write this equations as 
\begin{align}
\max_{x,t}~t ,  
\end{align}
\begin{align}sucht~that~
x^{H}(A_1-t B_1)x \geq 0
\end{align}
\begin{align}
x^{H}(A_2-t B_2)x \geq 0  
\end{align}
May some one here can come up with a algorithm for this problem. 
 A: Here is a crude idea that might work (haven't thought too carefully about it).
Let $a=\lambda_{\min}(B_1^{-1}A_1)$ and $b=\lambda_{\min}(B_2^{-1}A_2)$. Then, for there to be a feasible solution to the 2nd formulation, the variable $t$ must lie in the interval $[0,t_{\max}]$, where $t_{\max} := \min(a,b)$. 
This suggests that maybe we can do a binary-search for the optimal value of $t$.


*

*Start with $t = t_{\max}/2$

*With this value of $t$, either there is a feasible $x$ that satisfies both inequalities, or there isn't. If there isn't, we shrink $t$, else we expand it; that is, if there is no feasible $x$, we search in $[0,t]$, otherwise we search in $[t, t_{\max}]$ and repeat this step.


(Notice that we obtain an optimal $x$ as a by-product).
A: Problem solved in (7) in
Jianshu Zhang; Roemer, F.; Haardt, M.; Khabbazibasmenj, A.; Vorobyov, S.A., "Sum rate maximization for multi-pair two-way relaying with single-antenna amplify and forward relays," Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on , vol., no., pp.2477,2480, 25-30 March 2012
doi: 10.1109/ICASSP.2012.6288418
