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First question in these forums so go easy on me. I have a function $f_i(x):\mathbb{R}^N\to\mathbb{R}$ which is defined by $f_i(x) = \frac{(x^TAe_i)x^TAx}{(x^TA+b^T){\bf 1}}$ where $x\in\mathbb{R}^N$, $b\in\mathbb{R}^N$, $A\in\mathbb{R}^{N\times N}$, $e_i$ is the indicator vector with a $1$ in the $i^{th}$ position, and ${\bf 1}$ is an $N$-dimensional column vector of ones.

I want to determine if $f_i(x)$ is convex in $x$. We are allowed to use the fact that $A$ is positive-semidefinite. Thus I know that $x^TAx$ is convex in $x$, but I'm not sure if this is useful. What is throwing me off the is the cubic dependence on $x$ in the numerator.

Any tips on how to proceed? Thanks!

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    Thanks Pavel, I'll post here if I come up with$a$reasonable result.2013-01-01

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