DE optimization with equality constraint

Reply to "Optimization with constraint" on March 14, 2008
One can get an accurate solutons applying the "Differential Evolution" algorithm
as implemented in the DEoptim package:

    f2 <- function(x){
        if (x[1] + x[2] < 1 || x[1] + x[2] > 1) {
            r <- Inf
        } else {
            r <- x[1]^2 + x[2]^2
        }
        return(r)
    }

    lower <- c(0, 0)
    upper <- c(1, 1)

    DEoptim(f2, lower, upper, control=list(refresh=200))$bestmem

    iteration:  200 best member:  0.5 0.5 best value:  0.5

This approach assumes nothing about the gradient, hessian or whatever. And the
equality is split into two inequalities assuming no relaxation or penalty.
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?

Again, the simple example should only show me the
basic working of the complex functions in R.

Hope you can help me.

With regards
Andreas.
Reply to "Optimization with constraint" on March 14, 2008
 One can get an accurate solutons applying the "Differential Evolution" algorithm
 as implemented in the DEoptim package:

    f2 <- function(x){
        if (x[1] + x[2] < 1 || x[1] + x[2] > 1) {
            r <- Inf
        } else {
            r <- x[1]^2 + x[2]^2
        }
        return(r)
    }

    lower <- c(0, 0)
    upper <- c(1, 1)

    DEoptim(f2, lower, upper, control=list(refresh=200))$bestmem

    iteration:  200 best member:  0.5 0.5 best value:  0.5

 This approach assumes nothing about the gradient, hessian or whatever. And the
 equality is split into two inequalities assuming no relaxation or penalty.
The problem with DEoptim approach is that is not guaranteed that it
converges to the solution. Moreover, from my experience, it seems to
be quite slow when the optimization problem is high-dimensional (i.e.,
with many variables).

Paul
 Andreas Klein <klein82517 <at> yahoo.de> wrote:
 >
 > 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?
 >
 > Again, the simple example should only show me the
 > basic working of the complex functions in R.
 >
 > Hope you can help me.
 >
 > With regards
 > Andreas.

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Paul Smith <phhs80 <at> gmail.com> writes:
The problem with DEoptim approach is that is not guaranteed that it
converges to the solution. Moreover, from my experience, it seems to
be quite slow when the optimization problem is high-dimensional (i.e.,
with many variables).

Paul
There is a difference between local and global optimization:

'optim' realizes *local* optimization using a gradient-based approach.
This is fast, but will get stuck in local optima (except method SANN). 
'DEoptim' is one of many approaches to *global* optimization, of which
each has its advantages and drawbacks.
...not guaranteed that it converges to the solution.
As a local optimization routine, also 'optim' does not guarantee to
  reach a (global) optimum.
[DEoptim] seems to be quite slow...
This is normal for routines in global optimization as they have to
  search a quite large space.

Hans Werner
 > The problem with DEoptim approach is that is not guaranteed that it
 > converges to the solution. Moreover, from my experience, it seems to
 > be quite slow when the optimization problem is high-dimensional (i.e.,
 > with many variables).
 There is a difference between local and global optimization:

 'optim' realizes *local* optimization using a gradient-based approach.
 This is fast, but will get stuck in local optima (except method SANN).
 'DEoptim' is one of many approaches to *global* optimization, of which
 each has its advantages and drawbacks.
Regarding this point, I agree with you.
 > ...not guaranteed that it converges to the solution.
  As a local optimization routine, also 'optim' does not guarantee to
  reach a (global) optimum.
Yes, but with optim one can be (almost) sure that the solution
returned is an optimum (at least locally); with DEoptim one cannot be
sure about that (it may be a non-optimum, locally or globally).
 > [DEoptim] seems to be quite slow...
  This is normal for routines in global optimization as they have to
  search a quite large space.
Surely, but just try to solve an optimization problem with 100
variables with both approaches. In spite of being the same problem,
you will probably see that optim is far faster than DEoptim.

Paul