[R] Optimization with constraint.

[Hans W Borchers]

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.