I am trying to minimize this function over x. If it is convex -- there should be no problem. But I doubt that it is convex. Any algorithmic thoughts?
$$F(\mathbf{x})= \frac{|\mathbf{a}^T\mathbf{\Sigma}\mathbf{x}|}{\mathbf{x}^T\mathbf{\Sigma}\mathbf{x}}$$
where $\mathbf{a},\, \mathbf{x}$ are real-valued vectors and $\mathbf{\Sigma}$ is a positive-definite matrix.
Clearly $F(x) \geq 0$. Since $\Sigma$ is positive definite, the system $$ \Sigma x = b $$
has a solution for any $b$. It then suffices to pick any $0 \neq b \in \mathrm{span}(a)^{\perp}$ and $x = \Sigma^{-1} b$, since $$ F(\Sigma^{-1} b) = \frac{\langle \Sigma a, \Sigma^{-1} b \rangle}{\|b\|^2_{\Sigma^{-1/2}}} = \frac{\langle a, b \rangle}{\|b\|_{\Sigma^{-1/2}}^2} = 0, $$ since $b \in \mathrm{span}(a)^{\perp} \implies \langle a, b \rangle = 0$.