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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?

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  • $\begingroup$ You say "maximizing" in title, and min in the display. Which one is it? $\endgroup$
    – Igor Rivin
    Commented Oct 21, 2016 at 8:33
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    $\begingroup$ One answer is to simply use Lagrange multipliers; in the end, this would mean solving a cubic equation in $\alpha$. Alternatively, an answer could be that you are looking "in the wrong space" for the answer - the proper vector space to look for an answer is $V \otimes V$, and you are looking in the diagonal subvariety (note: not even a vector subspace) of elements $v \otimes v$. $\endgroup$
    – user44191
    Commented Oct 21, 2016 at 9:45

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Here is a paper on minimizing products of positive definite forms 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.

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