How to solve this matrix optimization problem

Question

I am trying to solve this analytically - find w such that minimize $ \frac{w'\Omega_1 w}{w'\Omega_2 w}$ Here $ \Omega $ are covariance matrices so invertable and symmetric.

Any idea will be appreciated.

Answer 1

This is called the smallest generalized eigenvalue of the pair $(\Omega_1,\Omega_2)$, or, in your case, simply the smallest eigenvalue of $\Omega_2^{-1/2}\Omega_1\Omega_2^{-1/2}$ (perform the variable change $x = \Omega_2^{1/2}w$ and you have the definition of the smallest eigenvalue.