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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: www-optima.amp.i.kyoto-u.ac.jp/result/masterdoc/19zhang.pdf –  sonystarmap 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: mathoverflow.net/questions/76420/… ? –  sonystarmap Feb 21 '13 at 13:40
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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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