Optimization problem with ratio objective

Question

I need to solve the following optimization problem

$$ \text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad \|x\|_1=1 \quad (\text{or alternatively} \quad c^T|x|=1), $$

where $x \in \mathbb{R}^n$ is the optimization variable, $B \in \mathbb{R}^{n \times n}$ and $a,c \in \mathbb{R}^n$. The matrix $B$ is positive-semidefinite and symmetric, the vectors $a$ and $c$ are non-negative.

Can this optimization problem be transformed into a tractable quadratic programm? Which numerical optimization method is best suited for this problem?

Cheers R.

Answer 1

You can use trust region or line search methods to deal with nonlinear objective funstion.