I would like to find $x$ which minimizes the following equation:

$\frac{x^HAx}{x^HBx} + \frac{x^HCx}{x^HDx}$ where A, B, C, D are positive-definite. $x$ is not a very large vector (<1000 elements in size.)

The problem is easily solvable as a generalized eigenvalue problem when A or C=0, but I am not sure what the best approach is for the general case.

Thanks!