# [R] Multi-objective optimization

Peter Dalgaard p.dalgaard at biostat.ku.dk
Wed Oct 17 21:33:30 CEST 2007

```Alberto Monteiro wrote:
> Duncan Murdoch 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
>
>
Yes, but there's an alternative interpretation:

maximize g(x,y) over y  _subject to_  f(x,y) being maximized over x

=> maximize g(y,y) over y  => y = x = 1.5

(this sort of optimization problem arises naturally if you want a
least-squares fit to a function given by a partial differential equation
and solve the latter using some minimization method).

--
O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
(*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)                  FAX: (+45) 35327907

```