Date: Sep 7, 2013 3:40 PM
Author: RGVickson@shaw.ca
Subject: Re: An optimization problem

On Saturday, September 7, 2013 6:00:16 AM UTC-7, anal...@hotmail.com wrote:
> consider f(x1,x2,...xn) = {1 + sum(xi)^2)} / {1 +sum{x(i)}
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> 0 <=xi <=1 for all i.
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> The maximum function value of 1 occurs at either all x's = 0 or all x's = 1.
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> Can an explicit formula be given for the minimum?
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> Thanks.


The minimum occurs at x_1 = x_2 = ... = x_n = a. To see this, note that at the minimum, the sum in the denominator will have some value, say n*a, so you can look at the simper problem of
min (1 + sum(x_i^2,i=1..n)), subject to the constraints sum(x_i,i=1..n) = n*a and 0 <= x_i <= 1. If you drop the bound constraint (just keeping the sum constraint) you have a simple constrained problem for which the Lagrange multiplier method will work; it will give the global minimum because it is a convex optimization problem. You will find the solution to be that all x_i are equal to a.

Now your problem is to minimize (1+n*a^2)/(1+n*a), which you can do by simple calculus: a = [sqrt(1+n)-1]/n. Since this a lies between 0 and 1, all the constraints are obeyed.