AP
Posts:
137
Registered:
3/4/09


Re: An optimization problem
Posted:
Sep 9, 2013 2:51 AM


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+2a1=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^22a(1+sumx_i) =sum(x_ia)^2na^2+12a =sum(x_ia)^2>=0
and f has a mini for (a,a...a)

