
Re: An optimization problem
Posted:
Sep 9, 2013 4:32 AM


On Sunday, September 8, 2013 11:51:53 PM UTC7, AP wrote: > 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)
There is no need to check: for a given value of the sum(x_i) the function is strictly convex in (x_1,x_2,...,x_n), so a point that satisfies the Lagrangian conditions is a *global* minimum. That is: it is globally minimal to take all x_i = a and to then minimize over z.

