[R] another optimization question

[John Fox]

Dear R list members,

Since today seems to be the day for optimization questions, I have one that 
has been puzzling me:

I've been doing some work on sem, my structural-equation modelling package. 
The models that the sem function in this package fits are essentially 
parametrizations of the multinormal distribution.  The function uses optim 
and nlm sequentially to maximize a multinormal likelihood. One of the 
changes I've introduced is to use an analytic gradient rather than rely on 
numerical derivatives. (If I can figure it out, I'd like to use an analytic 
Hessian as well.)

I could provide additional details, but the question that I have is 
straightforward. I expected that using an analytic gradient would make the 
program faster and more stable. It *is* substantially faster, by up to an 
order of magnitude on the problems that I've tried. In one case, however, a 
model that converged (to the published solution) with numerical derivatives 
failed to converge with analytic derivatives. I can program around the 
problem, by having the program fall back to numerical derivatives when 
convergence fails, but I was surprised by this result, and I'm concerned 
that it reflects a programming problem or an error in my math. I suspect 
that if I had made such an error, however, the other examples I tried would 
not have worked so well.

So, my question is, is it possible in principle for an optimization to fail 
using a correct analytic gradient but to converge with a numerical 
gradient? If this is possible, is it a common occurrence?

Any help would be appreciated.

John

-----------------------------------------------------
John Fox
Department of Sociology
McMaster University
Hamilton, Ontario, Canada L8S 4M4
email: jfox at mcmaster.ca
phone: 905-525-9140x23604
web: www.socsci.mcmaster.ca/jfox
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