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The problem is $$\max f(\mathbf{x}) \text{ subject to } \mathbf{Ax} = \mathbf{b}$$

where $f(\mathbf{x}) = \sum_{i=1}^N\sqrt{1+\frac{x_i^4}{(\sum_{i=1}^{N}x_i^2)^2}}$,
$\mathbf{x} = [x_1,x_2,...,x_N]^T \in \mathbb{R}^{N\times 1}$, and
$\mathbf{A} \in \mathbb{R}^{M\times N} $

$f(.)$ is in the form of $\sqrt{1+y^2}$. Hence f(.) is a convex function. which is bounded in $[\sqrt{2}, 2]$.

I am looking for efficient algorithms which solve the above problem or convex maximization problem, in general.

Also, is it possible to solve the above problem using standard convex optimization tools by putting more constraints?

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Welcome to cstheory, a Q&A site for research-level questions in theoretical computer science (TCS). Your question does not appear to be a research-level question in TCS. Please see the FAQ for more information on what is meant by this. Your question might be suitable for Computer Science which has a broader scope. –  Kaveh Dec 24 '12 at 21:37
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ps: the question might be fine here if you edit it so it is not as localized, e.g. you can ask more generally about efficient algorithms for solving classes of convex maximization problems. –  Kaveh Dec 24 '12 at 21:39
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