Subject: Re: CCL:optimization with constraints

From: [email-masked] (Eric Martin)
Date: Thu, 23 Mar 95 12:48:47 -0800

Dear Antonio & Bert (CCL),

>>I am trying to do a non linear least squares fit,
>>using a Marquardt type algoritm, but I need to restrict some
>>parameters to make my function behaves correctly.
>>Is available any program to constrain parameters to be positive?
> 
> formulate your problem with transformed parameters, such that
> p_new = ln(p_old)
> optimize p_new (without constraints) and obtain your original parameter (p_old) as
> p_old = exp(p_new)
> 
> (this is just one of the many transformations that map {-inf,+inf}
> onto {0,+inf}.  Any other transformation that does the same might be
> used equally well. Sometimes the nature of the problem suggests a
> particular transformation.  The choice of transformation may affect the
> efficiency of the optimization algorithm) 

My understanding is that fitting a transformed function might not only
affect the efficiency, it can also affect the answer.  The least
squares best fit of the transformed problem will have different
parameter values (even after the reverse transorm) than the best fit of
the untransformed (albeit constrained) problem, since the sum of
squared residuals of a different function is being minimized.  This can
sometimes be rectified by applying weighted least squares.  See:
"When, Why and How to Use Weighted Least Squares", Robert de Levie, J.
Chem. Edu., 63, (1986) p. 10.

Best,  -Eric

Eric Martin
[email-masked]
4560 Horton St., Emeryville, CA 94608
(510)601-3306,  FAX (510)601-3360