Consider the following problem that is known to be non-convex but can be solved as a generalized eigenvalue problem (i.e. has a global optimum solution):
$\underset{w}{\text{maximize}}\quad w^{\top}Qw\\ \underset{w}{\text{minimize}}\quad w^{\top}Pw$
where $P\in\mathcal{R}^{n\times n}$ and $Q\in\mathcal{R}^{n\times n}$ are symmetric, PSD and their elements are NOT necessarily non-negative. Also, $w\in\mathcal{R}^{n}$.
We need to solve $\underset{w}{\text{maximize}}\:\frac{w^{\top}Qw}{w^{\top}Pw}$. This can be written as $Qw=\lambda Pw$ and if P is full rank we can convert this to a standard eigenvalue problem: $P^{-1}Qw=\lambda w$.
The problem that I want to solve is similar to above problem except in that the elements of vector $w$ must be non-negative:
$\underset{w}{\text{maximize}}\quad w^{\top}Qw\\ \underset{w}{\text{minimize}}\quad w^{\top}Pw$
$s.t.\qquad w\geq0$
or equivalently $\underset{w}{\text{maximize}}\:\frac{w^{\top}Qw}{w^{\top}Pw}$ s.t. $\: w\geq0$. What kind of method can be used to solve this?
