Constrined dependent optimization.

Apparently, the convergence is faster if one uses this new swap function:

swapfun <- function(x,N=100) {
 loc <- c(sample(1:(N/2),size=1,replace=FALSE),sample((N/2):100,1))
 tmp <- x[loc[1]]
 x[loc[1]] <- x[loc[2]]
 x[loc[2]] <- tmp
 x
}

It seems that within 20 millions of iterations, one gets the exact
optimal solution, which does not take too long.

Paul


On Mon, Mar 30, 2009 at 5:11 PM, Paul Smith <phhs80 at gmail.com> wrote:

Optim with SANN also solves your example:

-------------------------------------------

f <- function(x) sum(c(1:50,50:1)*x)

swapfun <- function(x,N=100) {
?loc <- sample(N,size=2,replace=FALSE)
?tmp <- x[loc[1]]
?x[loc[1]] <- x[loc[2]]
?x[loc[2]] <- tmp
?x
}

N <- 100

opt1 <- optim(fn=f,par=sample(1:N,N),gr=swapfun,method="SANN",control=list(maxit=50000,fnscale=-1,trace=10))
opt1$par
opt1$value

-------------------------------------------

We need to specify a large number of iterations to get the optimal
solution. The objective function at the optimum is 170425, and one
gets a close value with optim and SANN.

Paul


On Mon, Mar 30, 2009 at 2:22 PM, Hans W. Borchers
<hwborchers at googlemail.com> wrote:

Image you want to minimize the following linear function

? ?f <- function(x) sum( c(1:50, 50:1) * x / (50*51) )

on the set of all permutations of the numbers 1,..., 100.

I wonder how will you do that with lpSolve? I would simply order
the coefficients and then sort the numbers 1,...,100 accordingly.

I am also wondering how optim with "SANN" could be applied here.

As this is a problem in the area of discrete optimization resp.
constraint programming, I propose to use an appropriate program
here such as the free software Bprolog. I would be interested to
learn what others propose.

Of course, if we don't know anything about the function f then
it amounts to an exhaustive search on the 100! permutations --
probably not a feasible job.

Regards, ?Hans Werner



Paul Smith wrote:

On Sun, Mar 29, 2009 at 9:45 PM, ?<rkevinburton at charter.net> wrote:

I have an optimization question that I was hoping to get some suggestions
on how best to go about sovling it. I would think there is probably a
package that addresses this problem.

This is an ordering optimzation problem. Best to describe it with a
simple example. Say I have 100 "bins" each with a ball in it numbered
from 1 to 100. Each bin can only hold one ball. This optimization is that
I have a function 'f' that this array of bins and returns a number. The
number returned from f(1,2,3,4....) would return a different number from
that of f(2,1,3,4....). The optimization is finding the optimum order of
these balls so as to produce a minimum value from 'f'.I cannot use the
regular 'optim' algorithms because a) the values are discrete, and b) the
values are dependent ie. when the "variable" representing the bin
location is changed (in this example a new ball is put there) the
existing ball will need to be moved to another bin (probably swapping
positions), and c) each "variable" is constrained, in the example above
the only allowable values are integers from 1-100. So the problem becomes
finding the optimum order of the "balls".

Any suggestions?

If your function f is linear, then you can use lpSolve.

Paul

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