It seems that there is here a problem of reliability, as one never
knows whether the solution provided by R is correct or not. In the
case that I reported, it is fairly simple to see that the solution
provided by R (without any warning!) is incorrect, but, in general,
that is not so simple and one may take a wrong solution as a correct
one.

Paul


On 5/8/07, Ravi Varadhan <rvaradhan@(protected):
> Your function, (x1-x2)^2, has zero gradient at all the starting values such
> that x1 = x2, which means that the gradient-based search methods will
> terminate there because they have found a critical point, i.e. a point at
> which the gradient is zero (which can be a maximum or a minimum or a saddle
> point).
>
> However, I do not why optim converges to the boundary maximum, when analytic
> gradient is supplied (as shown by Sundar).
>
> Ravi.
>
> ----------------------------------------------------------------------------
> -------
>
> Ravi Varadhan, Ph.D.
>
> Assistant Professor, The Center on Aging and Health
>
> Division of Geriatric Medicine and Gerontology
>
> Johns Hopkins University
>
> Ph: (410) 502-2619
>
> Fax: (410) 614-9625
>
> Email: rvaradhan@(protected)
>
> Webpage: http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html
>
>
>
> ----------------------------------------------------------------------------
> --------
>
>
> -----Original Message-----
> From: r-help-bounces@(protected)
> [mailto:r-help-bounces@(protected)
> Sent: Monday, May 07, 2007 6:26 PM
> To: R-help
> Subject: Re: [R] Bad optimization solution
>
> On 5/7/07, Paul Smith <phhs80@(protected):
> > > I think the problem is the starting point. I do not remember the
> details
> > > of the BFGS method, but I am almost sure the (.5, .5) starting point is
> > > suspect, since the abs function is not differentiable at 0. If you
> perturb
> > > the starting point even slightly you will have no problem.
> > >
> > >         "Paul Smith"
> > >         <phhs80@(protected)
> > >         >
> To
> > >         Sent by:            R-help <r-help@(protected)>
> > >         r-help-bounces@(protected)
> cc
> > >         at.math.ethz.ch
> > >
> Subject
> > >                           [R] Bad optimization solution
> > >         05/07/2007 04:30
> > >         PM
> > >
> > >
> > >
> > >
> > >
> > >
> > >
> > >
> > > Dear All
> > >
> > > I am trying to perform the below optimization problem, but getting
> > > (0.5,0.5) as optimal solution, which is wrong; the correct solution
> > > should be (1,0) or (0,1).
> > >
> > > Am I doing something wrong? I am using R 2.5.0 on Fedora Core 6 (Linux).
> > >
> > > Thanks in advance,
> > >
> > > Paul
> > >
> > > ------------------------------------------------------
> > > myfunc <- function(x) {
> > >  x1 <- x[1]
> > >  x2 <- x[2]
> > >  abs(x1-x2)
> > > }
> > >
> > >
> optim(c(0.5,0.5),myfunc,lower=c(0,0),upper=c(1,1),method="L-BFGS-B",control=
> list(fnscale=-1))
> >
> > Yes, with (0.2,0.9), a correct solution comes out. However, how can
> > one be sure in general that the solution obtained by optim is correct?
> > In ?optim says:
> >
> >    Method '"L-BFGS-B"' is that of Byrd _et. al._ (1995) which allows
> >    _box constraints_, that is each variable can be given a lower
> >    and/or upper bound. The initial value must satisfy the
> >    constraints. This uses a limited-memory modification of the BFGS
> >    quasi-Newton method. If non-trivial bounds are supplied, this
> >    method will be selected, with a warning.
> >
> > which only demands that "the initial value must satisfy the constraints".
>
> Furthermore, X^2 is everywhere differentiable and notwithstanding the
> reported problem occurs with
>
> myfunc <- function(x) {
>  x1 <- x[1]
>  x2 <- x[2]
>  (x1-x2)^2
> }
>
> optim(c(0.2,0.2),myfunc,lower=c(0,0),upper=c(1,1),method="L-BFGS-B",control=
> list(fnscale=-1))
>
> Paul
>
> ______________________________________________
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>

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