Andreas Klein <klein82517 <at> yahoo.de>
writes:>
> Hello.
>
> I have some problems, when I try to model an
> optimization problem with some constraints.
>
> The original problem cannot be solved analytically, so
> I have to use routines like "Simulated Annealing" or
> "Sequential Quadric Programming".
>
> But to see how all this works in R, I would like to
> start with some simple problem to get to know the
> basics:
>
> The Problem:
> min f(x1,x2)= (x1)^2 + (x2)^2
> s.t. x1 + x2 = 1
>
> The analytical solution:
> x1 = 0.5
> x2 = 0.5
>
> Does someone have some suggestions how to model it in
> R with the given functions optim or constrOptim with
> respect to the routines "SANN" or "SQP" to obtain the
> analytical solutions numerically?
>
In optimization problems, very often you have to replace an equality by two
inequalities, that is you replace x1 + x2 = 1 with
min f(x1,x2)= (x1)^2 + (x2)^2
s.t. x1 + x2 >= 1
x1 + x2 <= 1
The problem with your example is that there is no 'interior' starting
point for
this formulation while the documentation for constrOptim requests:
The starting value must be in the interior of the feasible region,
but the minimum may be on the boundary.
You can 'relax', e.g., the second inequality with x1 + x2 <= 1.0001
and use
(1.00005, 0.0) as starting point, and you will get a solution:
>>> A <- matrix(c(1, 1, -1, -1), 2)
>>> b <- c(1, -1.0001)
>>> fr <- function (x) { x1 <- x[1]; x2 <- x[2]; x1^2 + x2^2 }
>>> constrOptim(c(1.00005, 0.0), fr, NULL, ui=t(A), ci=b)
$par
[1] 0.5000232 0.4999768
$value
[1] 0.5
[...]
$barrier.value
[1] 9.21047e-08
where the accuracy of the solution is certainly not excellent, but the solution
is correctly fulfilling x1 + x2 = 1.