[R] Can R solve this optimization problem?

[Paul Smith]

On Jan 7, 2008 1:04 AM, Duncan Murdoch <murdoch at stats.uwo.ca> wrote:
> >>> 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.
> >
> > Thanks, Duncan. I have placed a similar post in the Maxima list and
> > another one in the Octave list. (I have never used splines; so I did
> > not quite understand the method that you suggested to me.)
>
> Linear splines are just piecewise linear functions.  An easy way to
> parametrize them is by their value at a sequence of locations; they
> interpolate linearly between there.
>
> x would be piecewise quadratic, so its integral would be a sum of cubic
> terms.

Thanks, Duncan, for your explanation.

Paul