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From: "George A." <ga2311@columbia.edu>
Newsgroups: comp.soft-sys.matlab
Subject: Re: First order optimality Measure
Date: Mon, 4 Feb 2013 21:36:11 +0000 (UTC)
Organization: Massachusetts Institute of Technology
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"George A." <ga2311@columbia.edu> wrote in message <kencav$mhu$1@newscl01ah.mathworks.com>...
> "Matt J" wrote in message <kehaap$qts$1@newscl01ah.mathworks.com>...
> > "George A." <ga2311@columbia.edu> wrote in message <kee8r0$mka$1@newscl01ah.mathworks.com>...
> > >
> > > Thank you for this Matt.
> > > No the bounds are the same in both cases [0,inf], and the tolerances as well
> > ===============
> > 
> > OK. Well, it could be happening for purely mathematical reasons. Consider even the very simple linear scalar lsq problem
> > 
> >  min f(x) = (a*x)^2/2
> > 
> > where a=1e5. Then at x0=1e-8, the objective function is quite small
> > 
> >  f(x0) = 5e-7
> > 
> > but the first order optimality measure, i.e. the derivative, is 100....considerably larger.
> ================
> 
> Well, I don't think this should be the case here, because I already have a point found as a solution for the first formulation (norm and 1st order optimality measure < 10^-6), but this same point when provided as initial point to the second (equivalent) formulation gives a 1st order optimality measure of 10^4..

Let me rephrase my question:
Since I have 2 formulations of the same problem, which are equivalent in algebraic terms. Is there any possible explanation (scaling problem of one of the 2 formulations, significant non-linearities or anything) why the numerical solution given to one, is not recognized as a (numerical) solution to the other?  
In other words could it be the case that equivalence in algebraic terms, does not necessarily mean equivalence in numerical terms? Or this is absurd and the only explanation can be an error with my code in one of the 2 versions (although I have not managed to find such an error up to now)