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Topic: An optimization problem
Replies: 19   Last Post: Sep 14, 2013 9:44 AM

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AP

Posts: 137
Registered: 3/4/09
Re: An optimization problem
Posted: Sep 9, 2013 2:51 AM
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On Sat, 7 Sep 2013 06:00:16 -0700 (PDT), analyst41@hotmail.com wrote:

>
>consider f(x1,x2,...xn) = {1 + sum(xi)^2)} / {1 +sum{x(i)}
>
>0 <=xi <=1 for all i.
>
>The maximum function value of 1 occurs at either all x's = 0 or all x's = 1.
>
>Can an explicit formula be given for the minimum?
>
>Thanks.

if minimum (or maximum) for x1,x2,...xn then
df/dx_i(x1,x2,...xn)=0

so 2x_i(1+sumx_i)-(1+sumx_i^2)=0 and x_i=a with
2a=(1+na^2)/(1+na)
na^2+2a-1=0
and because 0<=a<=1, a=(sqrt(n+1)-1)/n

but one must see if there is min or max

f(a,a..,a)=(1+na^2)/(1+na)=2a

the sign of f(x1,x2,...xn)-f(a,a,..,a)
is the sign of
1+sumx_i^2-2a(1+sumx_i)
=sum(x_i-a)^2-na^2+1-2a
=sum(x_i-a)^2>=0

and f has a mini for (a,a...a)



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