[R] Non-Linear Optimization - Query

[Paul Smith]

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
>>
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>
>