Test step sizes and parameter scaling for LSQNONLIN (and the Levenberg Marquardt algorithm)

I have a multi-parameter and multi-objective optimisation problem for which I am trying to use LSQNONLIN.

1) On its initial iterations it appears that LSQNONLIN attempts to vary the parameters by as little as 1e-8. Is this correct? E.g. for parameters [1 1 1] it will investigate [1+1e-8 1 1] and then [1 1+1e-8 1} etc. For my problem these step-sizes hardly make a difference, thus wasting a lot of time, or causing it to make a poor initial evaluation. Is there any way to increase/control this step size?

2) What is the best way to scale the parameters. In my case I have parameters having an effect like p1*x^p2 so p1 has a linear effect while p2 causes a lot of non-linearity. At the moment I was scaling the parameters to all equal 1 i.e. normalize them. However in reality they are actually say p1=1e-4 and p2=12.

3) If I normalize the initial parameter set P by dividing by P itself I obtain ones (provided the initial set does not contain zeros). If I instead divide by (P*10) or (P*100) or (P*1000) I am effectively increasing the effect of the small step-size 1e-8. Is this a good approach to simply scale the initial parameter set closer to the step-size?

Thanks,

Kevin

Answer 1

[Alan Weiss]

The initial perturbations are lsqnonlin attempting to estimate the Jacobian of the objective. You can control the gradient estimation as explained here.

Good luck,

Alan Weiss

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