Consider the function $\ f(x,y,z,\rho_a,\rho_b)=$
$ \log \left(1+ (x+ \rho_ay)^2 + \frac{(z+ \rho_by)^2}{1+(x+ \rho_by)^2} \right)+ \log\left(1+ (x+ \rho_by)^2 + \frac{(z+ \rho_ay)^2}{1+(x+ \rho_ay)^2} \right)$
For the given function above, I want to optimize over $\rho_a$ and $\rho_b$ and after I find the optimal answers, I would like to evaluate the function above.
It might not be easy to evaluate the above, however can anyone think of anything smart to take advantage of the symmetry between the two expressions within $f$?
I am guessing that $\rho_a= \rho_b$?
Thanks