Re: [R] optimization problem in R ... can this be done?

From: Gregory Gentlemen <gregory_gentlemen_at_yahoo.ca>
Date: Sun 26 Jun 2005 - 09:26:15 EST


Spencer: Thank you for the helpful suggestions. I have another question following some code I wrote. The function below gives a crude approximation for the x of interest (that value of x such that g(x,n) is less than 0 for all n).  

# // btilda optimize g(n,x) for some fixed x, and then approximately finds that g(n,x) # such that abs(g(n*,x)=0 // btilda <- function(range,len) {
# range: over which to look for x
bb <- seq(range[1],range[2],length=len)
OBJ <- sapply(bb,function(x) {fixed <- c(x,1000000,692,500000,1600,v1,217227); return(optimize(g,c(1,1000),maximum=TRUE,tol=0.0000001,x=fixed)$objective)}) tt <- data.frame(b=bb,obj=OBJ)
tt$absobj <- abs(tt$obj)
d <- tt[order(tt$absobj),][1:3,]
return(as.vector(d))
}  

For instance a run of
> btilda(c(20.55806,20.55816),10000) .... returns:

               b                  obj                    absobj
5834   20.55812    -0.0004942848    0.0004942848
5833   20.55812     0.0011715433    0.0011715433
5835 20.55812 -0.0021601140 0.0021601140

My question is how to improve the precision of b (which is x) here. It seems that seq(20.55806,20.55816,length=10000 ) is only precise to 5 or so digits, and thus, is equivalent for numerous succesive values. How can I get around this?

Spencer Graves <spencer.graves@pdf.com> wrote: Part of the R culture is a statement by Simon Blomberg immortalized in library(fortunes) as, "This is R. There is no if. Only how."

I can't see now how I would automate a complete solution to your problem in general. However, given a specific g(x, n), I would start by writing a function to use "expand.grid" and "contour" to make a contour plot of g(x, n) over specified ranges for x = seq(0, x.max, length=npts) and n = seq(0, n.max, npts) for a specified number of points npts. Then I'd play with x.max, n.max, and npts until I got what I wanted. With the right choices for x.max, n.max, and npts, the solution will be obvious from the plot. In some cases, nothing more will be required.

If I wanted more than that, I would need to exploit further some specifics of the problem. For that, permit me to restate some of what I think I understood of your specific problem:

(1) For fixed n, g(x, n) is monotonically decreasing in x>0.

(2) For fixed x, g(x, n) has only two local maxima, one at n=0 (or n=eps>0, esp arbitrarily small) and the other at n2(x), say, with a local minimum in between at n1(x), say.

With this, I would write functions to find n1(x) and n2(x) given x. I might not even need n1(x) if I could figure out how to obtain n2(x) without it. Then I'd make a plot with two lines (using "plot" and "lines") of g(x, 0) and g(x, n2(x)) vs. x.

By the time I'd done all that, if I still needed more, I'd probably have ideas about what else to do.

hope this helps.
spencer graves

Gregory Gentlemen wrote:

> Im trying to ascertain whether or not the facilities of R are sufficient for solving an optimization problem I've come accross. Because of my limited experience with R, I would greatly appreciate some feedback from more frequent users.
> The problem can be delineated as such:
>
> A utility function, we shall call g is a function of x, n ... g(x,n). g has the properties: n > 0, x lies on the real line. g may take values along the real line. g is such that g(x,n)=g(-x,n). g is a decreasing function of x for any n; for fixed x, g(x,n) is smooth and intially decreases upon reaching an inflection point, thereafter increasing until it reaches a maxima and then declinces (neither concave nor convex).
>
> My optimization problem is to find the largest positive x such that g(x,n) is less than zero for all n. In fact, because of the symmetry of g around x, we need only consider x > 0. Such an x does exists in this problem, and of course g obtains a maximum value of 0 at some n for this value of x. my issue is writing some code to systematically obtain this value.
>
> Is R capable of handling such a problem? (i.e. through some sort of optimization fucntion, or some sort of grid search with the relevant constraints)
>
> Any suggestions would be appreciated.
>
> Gregory Gentlemen
> gregory_gentlemen@yahoo.ca
>
>
>
> The following is a sketch of an optimization problem I need to solve.
>
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Spencer Graves, PhD
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Received on Sun Jun 26 09:29:02 2005

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