Hi,
The absolute function stopping criterion is not meant for any positive
objective function. It is meant for functions whose minimum is 0. Here is
what David Gay's documentation from PORT says:
"6 - absolute function convergence: |f (x)| < V(AFCTOL) = V(31). This test is
only of interest in
problems where f (x) = 0 means a ‘‘perfect fit’’, such as nonlinear
least-squares problems."
For example, let us try a positive objective function:
> nlminb( obj = function(x) x^2 + 1, start=1, lower=-Inf, upper=Inf,
> control=list(trace=TRUE))
0: 2.0000000: 1.00000
1: 1.0000000: 0.00000
2: 1.0000000: 0.00000
$par
[1] 0
$objective
[1] 1
$convergence
[1] 0
$message
[1] "relative convergence (4)"
$iterations
[1] 2
$evaluations
function gradient
3 2
Here the absolute function criterion does not kicks in.
Now let us try a function whose minimum value is 0.
> nlminb( obj = function(x) x^2, start=6, grad=function(x) 2*x, lower=-Inf,
> upper=Inf, control=list(trace=TRUE) )
0: 36.000000: 6.00000
1: 4.0000000: 2.00000
2: 4.9303807e-32: 2.22045e-16
$par
[1] 2.220446e-16
$objective
[1] 4.930381e-32
$convergence
[1] 0
$message
[1] "absolute function convergence (6)"
$iterations
[1] 2
$evaluations
function gradient
4 3
We see that convergence is attained and that the stoppage is due to absolute
function criterion.
Suppose, we now set abs.tol=0:
> nlminb( obj = function(x) x^2, start=6, grad=function(x) 2*x, lower=-Inf,
> upper=Inf, control=list(trace=TRUE, abs.tol=0) )
0: 36.000000: 6.00000
1: 4.0000000: 2.00000
2: 4.9303807e-32: 2.22045e-16
3: 2.4308653e-63: -4.93038e-32
4: 2.9962729e-95: -5.47382e-48
5:1.4772766e-126: 1.21543e-63
6:1.8208840e-158: 1.34940e-79
7:8.9776511e-190: -2.99627e-95
8:1.1065809e-221: -3.32653e-111
9:5.4558652e-253: 7.38638e-127
10:6.7248731e-285: 8.20053e-143
11:3.3156184e-316: -1.82088e-158
12: 0.0000000: -2.02159e-174
13: 0.0000000: -2.02159e-174
$par
[1] -2.021587e-174
$objective
[1] 0
$convergence
[1] 0
$message
[1] "X-convergence (3)"
$iterations
[1] 13
$evaluations
function gradient
15 13
>
Now, we see that it takes a while to stop, eventhough it is clear that
convergence has been attained after 2 iterations. This demonstrates the need
for the absolute function criterion for obj functions whose minimum is exactly
0. Although, there is nothing wrong with setting abs.tol=0, except for some
loss of computational efficiency.
Now, let us get back to Matthew' example:
> nlminb( obj = function(x) x, start=1, lower=-2, upper=2,
> control=list(trace=TRUE))
0: 1.0000000: 1.00000
1: 0.0000000: 0.00000
$par
[1] 0
$objective
[1] 0
$convergence
[1] 0
$message
[1] "absolute function convergence (6)"
$iterations
[1] 1
$evaluations
function gradient
2 2
> nlminb( obj = function(x) x, start=1, lower=-2, upper=2,
> control=list(trace=TRUE, abs.tol=0))
0: 1.0000000: 1.00000
1: 0.0000000: 0.00000
2: -2.0000000: -2.00000
3: -2.0000000: -2.00000
$par
[1] -2
$objective
[1] -2
$convergence
[1] 0
$message
[1] "both X-convergence and relative convergence (5)"
$iterations
[1] 3
$evaluations
function gradient
3 3
Thus it is evident that setting abs.tol=0 is a reasonable, general solution for
functions whose minimum value is non-zero, because it protects against
premature termination at iteration `n' whenever |f(x_n)| < abs.tol. The only
limitation being that of loss of efficiency in problems where f(x*) = 0. where
x* is the local minimum.
Ravi.
____________________________________________________________________
Ravi Varadhan, Ph.D.
Assistant Professor,
Division of Geriatric Medicine and Gerontology
School of Medicine
Johns Hopkins University
Ph. (410) 502-2619
email: rvarad...@jhmi.edu
----- Original Message -----
From: Duncan Murdoch <murdoch.dun...@gmail.com>
Date: Friday, July 9, 2010 6:54 pm
Subject: Re: [R] Not nice behaviour of nlminb (windows 32 bit, version 2.11.1)
To: Matthew Killeya <matthewkill...@googlemail.com>
Cc: Peter Ehlers <ehl...@ucalgary.ca>, Ravi Varadhan <rvarad...@jhmi.edu>,
r-help@r-project.org, ba...@stat.wisc.edu
> On 09/07/2010 6:09 PM, Matthew Killeya wrote:
> >Yes clearly a bug... there are numerous variations ... problem seems
> to be
> >for a linear function whenever the first function valuation is 1.
> >
>
> Not at all. You can get the same problem on a quadratic that happens
> to have a zero at an inconvenient place, e.g.
>
> nlminb( obj = function(x) x^2-25, start=6, lower=-Inf, upper=Inf )
>
> Ravi's workaround of setting the abs.tol to zero fixes this example,
> but I think it's pretty clear from the documentation that the whole
> thing was designed for positive objective functions, so I wouldn't
> count on his workaround solving all the problems.
>
> Duncan Murdoch
>
>
> >e.g. two more examples:
> > nlminb( obj = function(x) x+1, start=0, lower=-Inf, upper=Inf )
> > nlminb( obj = function(x) x+2, start=-1, lower=-Inf, upper=Inf )
> >
> >(I wasn't sure where best to report a bug, so emailed the help list)
> >
> >On 9 July 2010 22:10, Peter Ehlers <ehl...@ucalgary.ca> wrote:
> >
> >
> >>Actually, it looks like any value other than 1.0
> >>(and in (lower, upper)) for start will work.
> >>
> >> -Peter Ehlers
> >>
> >>
> >>On 2010-07-09 14:45, Ravi Varadhan wrote:
> >>
> >>
> >>>Setting abs.tol = 0 works! This turns-off the absolute function
> >>>convergence
> >>>criterion.
> >>>
> >>>
> >>> nlminb( objective=function(x) x, start=1, lower=-2, upper=2,
> >>> control=list(abs.tol=0))
> >>>$par
> >>>[1] -2
> >>>
> >>>$objective
> >>>[1] -2
> >>>
> >>>$convergence
> >>>[1] 0
> >>>
> >>>$message
> >>>[1] "both X-convergence and relative convergence (5)"
> >>>
> >>>$iterations
> >>>[1] 3
> >>>
> >>>$evaluations
> >>>function gradient
> >>> 3 3
> >>>
> >>>
> >>>This is clearly a bug.
> >>>
> >>>
> >>>Ravi.
> >>>
> >>>-----Original Message-----
> >>>From: r-help-boun...@r-project.org [
> >>>On
> >>>Behalf Of Ravi Varadhan
> >>>Sent: Friday, July 09, 2010 4:42 PM
> >>>To: 'Duncan Murdoch'; 'Matthew Killeya'
> >>>Cc: r-help@r-project.org; ba...@stat.wisc.edu
> >>>Subject: Re: [R] Not nice behaviour of nlminb (windows 32 bit, version
> >>>2.11.1)
> >>>
> >>>Duncan, `nlminb' is not intended for non-negative functions only.
> There
> >>>is
> >>>indeed something strange happening in the algorithm!
> >>>
> >>>start<- 1.0 # converges to wrong minimum
> >>>
> >>>startp<- 1.0 + .Machine$double.eps # correct
> >>>
> >>>startm<- 1.0 - .Machine$double.eps # correct
> >>>
> >>> nlminb( objective=obj, start=start, lower=-2, upper=2)
> >>> $par
> >>>[1] 0
> >>>
> >>>$objective
> >>>[1] 0
> >>>
> >>>$convergence
> >>>[1] 0
> >>>
> >>>$message
> >>>[1] "absolute function convergence (6)"
> >>>
> >>>$iterations
> >>>[1] 1
> >>>
> >>>$evaluations
> >>>function gradient
> >>> 2 2
> >>>
> >>>
> >>>
> >>>>nlminb( objective=obj, start=startp, lower=-2, upper=2)
> >>>>
> >>>>
> >>>$par
> >>>[1] -2
> >>>
> >>>$objective
> >>>[1] -2
> >>>
> >>>$convergence
> >>>[1] 0
> >>>
> >>>$message
> >>>[1] "both X-convergence and relative convergence (5)"
> >>>
> >>>$iterations
> >>>[1] 3
> >>>
> >>>$evaluations
> >>>function gradient
> >>> 3 3
> >>>
> >>>
> >>>
> >>>>nlminb( objective=obj, start=startm, lower=-2, upper=2)
> >>>>
> >>>>
> >>>$par
> >>>[1] -2
> >>>
> >>>$objective
> >>>[1] -2
> >>>
> >>>$convergence
> >>>[1] 0
> >>>
> >>>$message
> >>>[1] "both X-convergence and relative convergence (5)"
> >>>
> >>>$iterations
> >>>[1] 3
> >>>
> >>>$evaluations
> >>>function gradient
> >>> 3 3
> >>>
> >>>
> >>> From the convergence message the `absolute function convergence'
> seems to
> >>> be
> >>>the culprit, although I do not understand why that stopping
> criterion is
> >>>becoming effective, when the algorithm is started at x=1, but not
> at any
> >>>other values. The documentation in IPORT makes it clear that this
> >>>criterion
> >>>is effective only for functions where f(x*) = 0, where x* is a local
> >>>minimum. In this example, x=0 is not a local minimum for f(x), so
> that
> >>>criterion should not apply.
> >>>
> >>>
> >>>Ravi.
> >>>
> >>>
> >>>-----Original Message-----
> >>>From: r-help-boun...@r-project.org [
> >>>On
> >>>Behalf Of Duncan Murdoch
> >>>Sent: Friday, July 09, 2010 3:45 PM
> >>>To: Matthew Killeya
> >>>Cc: r-help@r-project.org; ba...@stat.wisc.edu
> >>>Subject: Re: [R] Not nice behaviour of nlminb (windows 32 bit, version
> >>>2.11.1)
> >>>
> >>>On 09/07/2010 10:37 AM, Matthew Killeya wrote:
> >>>
> >>>
> >>>> nlminb( obj = function(x) x, start=1, lower=-Inf, upper=Inf )
> >>>>
> >>>>
> >>>>
> >>>If you read the PORT documentation carefully, you'll see that their
> >>>convergence criteria are aimed at minimizing positive functions.
> (They
> >>>never state this explicitly, as far as I can see.) So one stopping
> >>>criterion is that |f(x)|< abs.tol, and that's what it found for
> you. I
> >>>don't know if there's a way to turn this off.
> >>>
> >>>Doug or Deepayan, do you know if nlminb can be made to work on functions
> >>>that go negative?
> >>>
> >>>Duncan Murdoch
> >>>
> >>> $par
> >>>
> >>>>[1] 0
> >>>>
> >>>>$objective
> >>>>[1] 0
> >>>>
> >>>>$convergence
> >>>>[1] 0
> >>>>
> >>>>$message
> >>>>[1] "absolute function convergence (6)"
> >>>>
> >>>>$iterations
> >>>>[1] 1
> >>>>
> >>>>$evaluations
> >>>>function gradient
> >>>> 2 2
> >>>>
> >>>> [[alternative HTML version deleted]]
> >>>>
> >>>>
> >>>>
> >
> >
>
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