[Rd] "wild" function example in optim

[Prof Brian Ripley]

On Tue, 26 Apr 2005, Thomas Lumley wrote:

> On Tue, 26 Apr 2005, Werner Bier wrote:
>
>> Dear all,
>> 
>> Firstly, I do apologize if my question is simple and posted in the wrong 
>> place but I had no reply from the R-help mailing list (maybe it is too 
>> simple!).
>> 
>> I was wondering why parscale is set to 20 in the "wild" function example 
>> used in ?optim. This function has only one parameter and if we set parscale 
>> equal to 1 then the solution near the global minimum is not found.

Note that there the method is "SANN".  That makes assumptions about step 
sizes, in fact using a spherical Gaussian distribution of fixed size.  So 
parscale=20 is telling it to make initial steps large enough to explore 
the `blobs'. In particular, parscale is not set for the BFGS call in that 
example.

>> I would use parscale only in cases the object function has more than one 
>> parameter to be optimised, shouldn't I?
>> 
>
> parscale is more important in cases with more than one parameter (and with 
> one parameter you could set fnscale instead of parscale to get the same 
> effect)

Not necessarily.  The finite-differencing is done in units rescaled by 
parscale.  So a unit change in a single parameter needs to be a 
reasonably-sized step.  One can always set fnscale and neps, but it is 
easier to set parscale.

> However, a sufficiently badly scaled one-d problem can still benefit from 
> fnscale or parscale.
>> f
> function(x) 1e-10*x^2
>> g
> function(x) 2e-10*x
>> optim(7,f,g,method="CG")$par
> [1] 7
>> optim(7,f,g,method="CG",control=list(parscale=1e5))$par
> [1] 1.209735e-14
>> optim(7,f,g,method="CG",control=list(fnscale=1e-10))$par
> [1] 1.673141e-15

but without g

> optim(7,f,method="CG",control=list(parscale=1e5))$par
[1] 1.209735e-14
> optim(7,f,method="CG",control=list(fnscale=1e-10))$par
[1] 1.997947e-11


-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
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