[R] Multi-objective optimization

[Peter Dalgaard]

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
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