[R] Multi-objective optimization

[Paul Smith]

On 10/17/07, Alberto Monteiro <albmont at centroin.com.br> wrote:
> >> Is there any package to do multi-objective optimization? For instance,
> >> consider the following problem:
> >>
> >> maximize f(x,y) in order to x
> >>
> >> and
> >>
> >> maximize g(x,y) in order to y,
> >>
> >> simultaneously, with x and y being the same both for f and g. Can R do
> >> it numerically?
> >
> > I don't think the problem is well posed.  For example, what's the
> > solution if f(x,y) = -(x-y)^2 and g(x,y) = -(x-2)^2-(y-1)^2?  The
> > first is maximized at x=y, the second at x=2, y=1, so in order to
> > choose a solution you need to specify what sort of tradeoff to use
> > to combine the two objectives.
> >
> I guess the problem was not well _defined_.
>
> I "interpreted" it as:
>
> maximize f(x,y) in order to x %means%
> (1) for every y, find x = f1(y) such that f(x,y) is max
>
> maximize g(x,y) in order to y %means%
> (2) for every x, find y = g1(x) such that g(x,y) is max
>
> simultaneously %means%
> (3) x = f1(y) and y = g1(x).
>
> So, for your example, we would have:
> (1) => x = y
> (2) => y = 1
> (3) => x = y = 1

I apologize for having been not sufficiently clear. Alberto's
interpretation corresponds to what I wanted to mean. Can R solve
numerically this sort of maximization problems?

Paul