[R] Constrained optimization

[David Khabie-Zeitoune]

It seems that you have a nonlinear constraint (rather than box
constraints). Provided your constraint is reasonably well behaved
(differentiable) you could still use optim on the Lagrangian, and then
run an "outer" optimisation to find the langrangian multiplier and slack
variable (that is, assuming you have an INequality constraint. Set z to
zero below if you just have an equality constraint). For example if your
original problem was:

Max_{x} f(x)
such that g(x) >= 0

You could instead try:
Max_{x;y;z} f(x) - y * (g(x) - z)
such that z >= 0

i.e. define a function L(x,l,z) = f(x) - y * (g(x) - z) 

Then feed L into optim and maximise it w.r.t: x (your original
variables), y (the Lagrange multiplier) and z (a slack variable).

Don't hold me to this as my optimisation theory is a little rusty...

-----Original Message-----
From: Martin Maechler [mailto:maechler at stat.math.ethz.ch] 
Sent: 17 June 2003 08:42
To: Charles.Annis at statisticalengineering.com
Cc: r-help at stat.math.ethz.ch
Subject: Re: [R] Constrained optimization


>>>>> "FCharles" == Charles Annis, P E
<Charles.Annis at statisticalengineering.com>
>>>>>     on Mon, 16 Jun 2003 13:19:50 -0400 writes:

    FCharles> Greetings, R-Wizards: I'm trying to find an
    FCharles> extremum subject to a nonlinear constraint.  (Yes,
    FCharles> I have perused the archives but have found nothing
    FCharles> positive.) The details of the problem are these:

    FCharles> In a paper published some years ago in
    FCharles> Technometrics, ("Confidence bands for cumulative
    FCharles> distribution functions of continuous random
    FCharles> variables" Technometrics, 25, 77-86. 1983), Cheng
    FCharles> and Iles describe an ingenious method for placing
    FCharles> confidence bounds on an entire cdf by defining the
    FCharles> likelihood ratio confidence "ellipse" for the
    FCharles> model parameters, and then traversing the
    FCharles> periphery and finding the most extreme values for
    FCharles> x, at a given F(x), such that the distribution
    FCharles> parameters reside on that confidence contour.
    FCharles> (Well, their method is more sophisticated than
    FCharles> that, but that's essentially how it works.)  I
    FCharles> implemented the thing in a spreadsheet 15 years
    FCharles> ago, and would like to do the same in R.  But
    FCharles> EXCEL's solver can find an extremum subject to a
    FCharles> constraint, and I haven't figured out how to get
    FCharles> nlm() to do that.

    FCharles> I would be grateful for any algorithmic
    FCharles> suggestions.

I'm astonished that your perusing of the archives and the on-line help
system didn't find 
	optim().

It has many methods, one of which working with box constraints. I'm
urging you to read help(optim) {and please read to the end even if it's
long}.

If you have non-box constraints, there are other tricks
(eg. continuous border penalizing) but it really depends on 
your application.

Martin Maechler <maechler at stat.math.ethz.ch>
http://stat.ethz.ch/~maechler/
Seminar fuer Statistik, ETH-Zentrum  LEO C16	Leonhardstr. 27
ETH (Federal Inst. Technology)	8092 Zurich	SWITZERLAND
phone: x-41-1-632-3408		fax: ...-1228			<><

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