[R] Optimization under an absolute value constraint

[Paul Smith]

On 9/7/07, Phil Xiang <pxxiang at yahoo.com> wrote:
> I need to optimize a multivariate function f(w, x, y, z, ...) under an absolute value constraint. For instance:
>
>     min { (2x+y) (w-z) }
>
> under the constraint:
>
>     |w| +  |x| + |y| + |z| = 1.0 .
>
> Is there any R function that does this? Thank you for your help!

I think that the minimum value of the function

f(x) :=  -2*x*(1-x), with 0 <= x <= 1

is also the minimum value of the objective function of your problem
(but correct me if I am wrong). Thus,

x	y	w	z
-0.5	0	0	-0.5
-0.5	0	0.1	-0.4
-0.5	0	0.3	-0.2
0.5	0	-0.5	0
-0.5	0	0.5	0
0.5	0	-0.4	0.1
0.5	0	-0.2	0.3
0.5	0	0	0.5

are all solutions for your problem.

Paul