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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.

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This could be promising, but I only took a quick look at it: – cmmndy Feb 8 '13 at 8:55
I seems that nonlinear fractional programming is a well-known problem . However, the sticking point here is the absolute value in the denominator. – Robinaut Feb 21 '13 at 12:58
Does this help, together with my first link:… ? – cmmndy Feb 21 '13 at 13:40
Do we at least know that $B$ is positive definite and the entries of $c$ are non-negative? – fedja Feb 25 '13 at 1:49
@fedja Yes, B is positive definite and c are non-negative. Furthermore the constraint |x|=1 can be substituted by c^T|x|=1. – Robinaut Feb 25 '13 at 9:04

1 Answer 1

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

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