[R] Can R solve this optimization problem?

[Gabor Grothendieck]

If its too large the the primary portion of the objective basically
just looks like roundoff error.  By the way there is also rdonlp2
which can handle constraints.  Not on CRAN -- try google.

On Jan 9, 2008 6:45 AM, Paul Smith <phhs80 at gmail.com> wrote:
>
> On Jan 8, 2008 2:58 AM, Gabor Grothendieck <ggrothendieck at gmail.com> wrote:
> > > Thanks again, Gabor. Apparently, something is missing in your
> > > approach, as I tried to apply it to the original problem (without sin)
> > > and the result seems not being correct::
> > >
> > > f <- function(z) {
> > >  a <- sum(-(999:1)*z)
> > >  return(a)
> > > }
> > >
> > > result <- optim(rep(0,999), f,
> > > NULL,lower=rep(-1/1000,999),upper=rep(1/1000,999),method="L-BFGS-B")
> > > b <- result$par
> > >
> > > x <- 0
> > >
> > > for (i in 1:999) {
> > >  x[i+1] <- b[i] + x[i]
> > > }
> > >
> > > The vector x contains the solution, but it does not correspond to the
> > > analytical solution. The analytical solution is
> > >
> > > x(t) = t, if t <= 1/2 and
> > >
> > > x(t) = 1 - t, if t >= 1/2.
> > >
> > > Am I missing something?
> > >
> >
> > No, but I was -- the x[n] = 0 constraint.  Add it through
> > a penalty term and try this:
> >
> > theta <- 1000
> > n <- 100
> > x <-  rep(1/n, n)
> > f <- function(x) sum(n:1 * x) - theta * sum(x)^2
> > optim(x, f, lower = rep(-1, n), upper = rep(1, n), method = "L-BFGS-B",
> >   control = list(maxit = 1000, fnscale = -1, lmm = 25))
> > plot(cumsum(res$par))
>
> Thanks, Gabor. Inspired by your code, I have written the following:
>
> theta <- 50000000
> n <- 1000
> x <-  rep(1/n, n)
> f <- function(x) sum(n:1 * x) - theta * sum(x)^2
> res <- optim(x, f, lower = rep(-1/n, n), upper = rep(1/n, n), method =
> "L-BFGS-B",control = list(maxit = 2000, fnscale = -1))
>
> b <- res$par
>
> x <- 0
>
> for (i in 1:n) {
>  x[i+1] <- b[i] + x[i]
> }
>
> plot(seq(0,1,by=1/n),x,type="l")
>
> to solve the same problem. It works fine unless one replaces
>
> theta <- 50000000
>
> by
>
> theta <- 1e100
>
> According to a book that I consulted meanwhile, the larger the value
> of theta the better the solution should be. Why does not it happen
> with my example? Any clues?
>
> Paul
>
>
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