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I'm trying to minimize the following:

$||Ax - b|| - c$

$A$ : $K \times M$ matrix
$x$ : $M \times N$ unknowns ($M$ $N$-dimensional vectors)
$b$ : $K$ $N$-dimensional vectors
$c$ : $K$ known euclideian distances in $R^N$

Can this be expressed as a linear least squares problem? How should I approach this?

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What is the difference between minimizing $||Ax-b||-c$ and $||Ax-b||$? If you just need to minimize $||Ax-b||$, this is equivalent to minimizing $||Ax-b||^2$, which is a 'linear' least squares problem. –  copper.hat May 4 '12 at 0:59
$c$ is variable, and not necessarily close to $0$, so I can't just skip it.. –  Babis May 4 '12 at 1:37
So it is an unknown, not a known? –  copper.hat May 4 '12 at 1:51
Actually, I don't understand, you can choose $c$ to be arbitrarily large and make the cost as small as you want. You need to elaborate a bit more. –  copper.hat May 4 '12 at 2:11
Ok just to give a 'visual' of the problem, imagine that the $x$ vector is a vector of circle centers, $b$ is all zeros and $c$ is the vector of 'desired' sums of radii. So the whole problem can be described as: arrange these circles so that they touch each other and intersect as little as possible. My problem is something like circle packing with uneven radii, but I know beforehand all the neighbours of each circle. –  Babis May 4 '12 at 13:21

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