Constrined dependent optimization.

Sorry I sent a description of the function I was trying to minimize 
but I must not have sent it to this group (and you). Hopefully with 
this clearer description of my problem you might have some 
suggestions.

It is basically a warehouse placement problem. You have a warehouse 
that has many items each placed in a certain bin (the "real"
warehouse has about 20,000 of these bins, hence the large number of 
variables that I want to input to optimize). Now assume that an order 
comes in for three items A, B, and C. In the worst case A will be on 
one end of the warehouse, B in the middle and C on the other end of 
the warehouse. The "work" involved in getting these items to fulfill 
this order is roughly proportional to the distance from A to B plus 
the distance from B to C (assuming the absolute positions are sorted). 
So the cost for fulfilling this order is this distance. In the ideal 
world A, B, and C would be right next to each other and the 
cost/distance would be minimized. So the function I want to minimize 
would be placing these 20,000 items in such a way so that the minimum 
"work" is involved in fulfilling the orders for the past month or two. 
Clearer?

I can see that I may need to cut back on the variables 20,000 is 
probably too many. Maybe I can take the top 1,000 or so. I just am not 
sure of the packages available what to reasonably expect. I would like 
this optimization to complete in a reasonable amount of time (less 
than a few days). I have heard that SANN is slower than other 
optimization methods but it does have the feature of supplying a 
"gradient" as you pointed out. Are there other packages out there that 
might be better suited to such a large scale optimizaiton?

Thanks again.

Kevin
---- Paul Smith <phhs80 at gmail.com> wrote: 

As I told you before, without knowing the definition of your 
function f, one cannot help much.

Paul


On Wed, Apr 1, 2009 at 3:15 PM,  <rkevinburton at charter.net> wrote:

Thank you I had not considered using "gradient" in this fashion.
Now as an add on question. You (an others) have suggested using 
SANN. Does your answer change if instead of 100 "variables" or 
bins there are 20,000? From the documentation L-BFGS-B is designed 
for a large number of variables. But maybe SANN can handle this as 
well.

Kevin

---- Paul Smith <phhs80 at gmail.com> wrote:

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=l
ist(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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