[R] Can R solve this optimization problem?

[Duncan Murdoch]

On 06/01/2008 7:05 PM, Paul Smith wrote:
> Dear All,
> 
> I am trying to solve the following maximization problem with R:
> 
> find x(t) (continuous) that maximizes the
> 
> integral of x(t) with t from 0 to 1,
> 
> subject to the constraints
> 
> dx/dt = u,
> 
> |u| <= 1,
> 
> x(0) = x(1) = 0.
> 
> The analytical solution can be obtained easily, but I am trying to
> understand whether R is able to solve numerically problems like this
> one. I have tried to find an approximate solution through
> discretization of the objective function but with no success so far.

R doesn't provide any way to do this directly.  If you really wanted to 
do it in R, you'd need to choose some finite dimensional parametrization 
of u (e.g. as a polynomial or spline, but the constraint on it would 
make the choice tricky:  maybe a linear spline?), then either evaluate 
the integral analytically or numerically to give your objective 
function.  Then there are some optimizers available, but in my 
experience they aren't very good on high dimensional problems:  so your 
solution would likely be quite crude.

I'd guess you'd be better off in Matlab, Octave, Maple or Mathematica 
with a problem like this.

Duncan Murdoch