Hi all, I have the following problem: I want to minimize Sum(a_i)^2
knowing that:
1. Sum(a_i)=N
2. all a_i are different

I know that removing condition 2., the solution is a_1 = a_2 = ... =
a_n = N/n. But what's the solution if I add condition 1.? Thanks!
--Ricardo

george seryakov

Hello!

rveloso@sapo.pt:

r> Hi all, I have the following problem: I want to minimize Sum(a_i)^2
r> knowing that:
r> 1. Sum(a_i)=N
r> 2. all a_i are different

r> I know that removing condition 2., the solution is a_1 = a_2 = ... =
r> a_n = N/n. But what's the solution if I add condition 1.?

Theoretically you can not minimize it. If you found a set of a_i it is
possible to find another with lesser Sum(a_i^2). Practically it could be
something as close to a_i = N/n as strict you want to fulfill the condition
2.

The != condition is not a well defined relation for real numbers, in
computational sense.

r> Thanks! --Ricardo
--
GS

helmut jarausch

A simple (brute force) technique would be to fix some strictly positive
numbers eps_i, 1<=i<=n-1, and reformulate the problem as
a_{i+1}= a_i+eps_i+dx_i
Then take a_1 and dx_i, 1 <= i <= n-1, as unknowns. Compute the minimum
under the constraints dx_i >= 0 .
Of course, the art is to fix the eps_i .


--
Helmut Jarausch

Lehrstuhl fuer Numerische Mathematik
RWTH - Aachen University
D 52056 Aachen, Germany

spellucci

In article <1105656275.451461.109390@c13g2000cwb.googlegroups .com>,
rveloso@sapo.pt writes:

you meant "add condition 2"

clearly there is no solution: you know the solution to the problem with
cond 1. now disturb the N/n by arbitrarily small numbers, then the sum becomes
larger but you might get arbitrarily near the original optimal value. hence there
is no minimizer for the problem with condition 2 added. (in more abstract terms
this reads: minimizing a continuous function on an open set is not well defined)
hth
peter