[R] Can R solve this optimization problem?

[Gabor Grothendieck]

On Jan 6, 2008 8:43 PM, Paul Smith <phhs80 at gmail.com> wrote:
>
> On Jan 7, 2008 1:32 AM, Gabor Grothendieck <ggrothendieck at gmail.com> wrote:
> > This can be discretized to a linear programming problem
> > so you can solve it with the lpSolve package.  Suppose
> > we have x0, x1, x2, ..., xn.  Our objective (up to a
> > multiple which does not matter) is:
> >
> > Maximize: x1 + ... + xn
> >
> > which is subject to the constraints:
> >
> > -1/n <= x1 - x0 <= 1/n
> > -1/n <= x2 - x1 <= 1/n
> > ...
> > -1/n <= xn - x[n-1] <= 1/n
> > and
> > x0 = xn = 0
> >
> >
> > On Jan 6, 2008 7:05 PM, Paul Smith <phhs80 at gmail.com> 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.
>
> Thats is clever, Gabor! But suppose that the objective function is
>
> integral of sin( x( t ) ) with t from 0 to 1
>
> and consider the same constraints. Can your method be adapted to get
> the solution?

If a linear approx is sufficient then yes; otherwise, no.  For
example, if x can
be constrained to be small then its roughly true that sin(x) = x and you are
back to the original problem.