I have run into a problem where I need to optimize (either minimize or maximize) - the following:
Given a symmetric positive definite $n \times n$ matrix $\Sigma$, to find an n-vector $x$, such that $|x|_2=1$, that either:
$$\max_x \mathrm{trace}\{x\Sigma x\} - \lambda~\mathrm{trace}\{x^{\top}\Sigma xx^{\top}\Sigma x\} $$
or:
$$\min_x \mathrm{trace}\{x\Sigma x\} - \lambda~\mathrm{trace}\{x^{\top}\Sigma xx^{\top}\Sigma x\} $$
I was wondering if anyone has encountered a problem like this and if there is any literature or results related to this ? Any pointers or ideas would be very helpful.
Thanks!