[R] Non-Linear Optimization - Query

[Ravi Varadhan]

Here is how you can implement the Lagrangian multiplier approach and solve
the first-order KKT conditions to obtain the solution for Paul Smith's
example:

	require(BB)

f2 <- function(x) {
f <- rep(NA, length(x))
f[1] <- 1 + 2 * x[1] * x[3]  # x[3] is the Lagrangian multiplier
f[2] <- 1 + 2 * x[2] * x[3]
f[3] <- x[1]^2 + x[2]^2 - 1  # the equality constraint
f
}

dfsane(par=rep(0, 3), fn=f2)
 

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


On Tue, Mar 17, 2009 at 11:33 AM, Lars Bishop <lars52r at gmail.com> wrote:
> Thanks Paul Sorry to ask this, but I'm new in R. Can't I just use the 
> Lagrangian as my objective function in BB? Otherwise, how do I 
> construct my system of equations?
>
> Thanks again
>
> Lars.
>
> On Mon, Mar 16, 2009 at 9:54 PM, Paul Smith <phhs80 at gmail.com> wrote:
>>
>> On Tue, Mar 17, 2009 at 12:09 AM, Lars Bishop <lars52r at gmail.com> wrote:
>> > I couple of weeks ago, I’ve asked for a package recommendation for 
>> > nonlinear optimization. In my problem I have a fairly complicated 
>> > non-linear objective function subject to one non-linear equality 
>> > constrain.
>> >
>> > I’ve been suggested to use the *Rdonlp2* package, but I did not get 
>> > any results after running the program for 5 hrs. Is it normal to 
>> > run this type of programs for hours? Also, I’d like to ask the 
>> > experts whether there is any other alternative I could use to solve 
>> > this. For example, can I define a Lagrange function (add lambda as 
>> > a parameter) and use optim() or any other optimization function?
>>
>> Are your objective function and your constraint differentiable? If 
>> so, then construct the Lagrangean and get the system of equations for 
>> calculating the first order conditions. This nonlinear system of 
>> equations can be solved with the package BB (by  Ravi Varadhan).
>>
>> Paul
>>
>> ______________________________________________
>> R-help at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>
>

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