Non-Linear Optimization - Query

Lars,

Here is how you can solve the example given by Paul using spg() in "BB"
package.

f <- function(x, lambda) x[1] + x[2] - lambda * (x[1]^2 + x[2]^2 - 1)^2  #
look at how Penalized function is formulated

eps <- Inf

tol <- 1.e-05

p0 <- c(1, 1)

lambda <- 0.1  # start with a small value for lambda

while (eps > tol) {

ans <- spg(fn=f, par=p0, lambda=lambda, control=list(maximize=TRUE,
trace=FALSE))

par <- ans$par

eps <- max(abs(p0 - par))

p0 <- par

lambda <- 10 * lambda  # increase the penalty

}

ans 

Hope this helps,
Ravi.

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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 at jhmi.edu

Webpage:  http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html

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-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On
Behalf Of Paul Smith
Sent: Tuesday, March 17, 2009 2:34 PM
To: r-help at r-project.org
Subject: Re: [R] Non-Linear Optimization - Query

Hi Lars,

Consider the following problem:

max x + y

subject to

x^2 + y^2 =1.

The solution is obviously (x,y) = (sqrt(2) / 2, sqrt(2) / 2).

Now, consider the unconstrained maximization problem on the variables x, y
and lambda:

max x + y + lambda * (x^2 + y^2 - 1)

(Notice that the objective function here corresponds to the Lagrangian.)

Clearly, this second problem has no maximum.

These two simple examples should make evident that your strategy of
maximizing the Lagrangian does not lead necessarily to a solution.

What do you mean by "how do I construct my system of equations"? Do you mean
how to derive analytically the equations? Or do you mean how to insert them
into R?

Best,

Paul

Non-Linear Optimization - Query

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