[Wien] BFGS Warning

[L. D. Marks]

On Wed, 22 Sep 2004, Jack London wrote:

> Dear all
>
> When I using PORT to minimize surface structure, it
> generate some warning message in case.outputM file.
>
>  :Warning: BFGS Theta reduced to  0.939262143119133
>
> What's the meaning of the warning message?
	A slightly technical explanation (for the record). Key to a BFGS
update is the curvature (or secant) condition:
	H s = y
where H is the current approximation to the Hessian, s is the change in
position and y the change in gradient (see, for instance,
http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/unconstrained/quasi.html
for more details).
	Dotting both sides on the left by s
	s H s = s.y
	The approximation to H should be positive definite (it's not the
real H), so s H s > 0. If s.y < 0, the curvature condition is said to
have failed so the standard BFGS update proceedure has to be modified. If
it's not modified, it will generate an H which is not positive definite
and everything goes wrong. In practice, a better method is to check that
	s.y > 0.2 (s H s)
If it is not, the update is damped (see for instance p12 of
http://www.math.ubc.ca/~loew/m604/web-ho/quasinew.pdf) using a parameter
"Theta". I added a warning to the code to let the user know if this is
happening.

Similar the the warning about the curvature condition, you should only
be concerned if this is happening repeatedly.
>
> Another question:
>
> I chose PORT method in case.inM not BFGS, why it
> generate BFGS warning message?
	Because PORT is a BFGS method! The old "BFGS" routine that is
still in Wien did not have the curvature condition traps discussed above
so tended to fail. It should probably be removed.
>
> case.inM
>
> PORT 5.0
> 0.6 0.6 0.6 1.0
> 0.6 0.6 0.6 1.0
> 0.6 0.6 0.6 1.0
> 0.6 0.6 0.6 1.0
> 0.0 0.0 0.0 1.0
> 0.6 0.6 0.6 1.0
>
> Best regards
>
> Zhang
>
>
>
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-----------------------------------------------
Laurence Marks
Department of Materials Science and Engineering
MSE Rm 2036 Cook Hall
2225 N Campus Drive
Northwestern University
Evanston, IL 60201, USA
Tel: (847) 491-3996 Fax: (847) 491-7820
mailto:L-marks at northwestern.edu
http://www.numis.northwestern.edu
-----------------------------------------------