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.