The problem is:


$$\min\_{\alpha}\frac{\alpha^T A \alpha}{\alpha^T\alpha}\frac{ \alpha^T B \alpha}{\alpha^T\alpha}$$


where $A$ and $B$ are symmetric and positive definite matrix.
I think the explicit solution may be hard to find. If anyone have some reference?



minimizing the product of rayleigh quotient
Here is a paper on [minimizing products of positive definite forms](http://epubs.siam.org/doi/pdf/10.1137/110821196) that you should find very useful --- it gives necessary and sufficient conditions for the products of positive definite quadratic forms to be convex, which will help do the optimization.


