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I've noticed the Levenberg-Marquardt algorithm is only used with least squares problem and I didn't find any library in R or Python which allow to minimise the absolute values of the residual (and not the square of the residual)

  1. Is there any reason of that ?
  2. Does it make sense to apply sqrt(abs(residual)) on the residual before it's been squared if we want to use this algorithm with Least Absolute Deviation instead of Least Squares Deviation ?
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  • $\begingroup$ Is it because sqrt(abs(x)) is not differentiable when x=0 ? $\endgroup$
    – psql
    Commented Oct 28, 2015 at 11:48

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Because it is based on a Second Order Approximation of the Squared Residual Function.

Hence it requires "Squared Residual Function".
The method will fit any model you can make which has the same form.

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