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

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Peter Percival

Posts: 2,623
Registered: 10/25/10
Re: An optimization problem
Posted: Sep 8, 2013 5:13 AM
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quasi wrote:
> William Elliot wrote:
>> 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?

>> Let g(x) = (1 + x^2)/(1 + x) and look for extreme values of g.

> The function g is not applicable.
> From the context, the OP intended
> f(x_1, ..., x_n) = (1 + sum((x_i)^2))/(1 + sum(x_i))
> whereas you interpreted it (incorrectly) as
> f(x_1, ..., x_n) = (1 + (sum(x_i))^2))/(1 + sum(x_i))
> Of the two possible interpretations above, only the first is
> consistent with the OP's claim that for all n, the maximum
> value of f is 1.
> quasi

I know nothing about such problems, but is it not clear that _if_ there
is a minimum, it occurs at some point where all the x_i's are equal?
Suppose that common value is x, then the minimum of

(1 + nx^2)/(1 + nx)

is sought. That occurs at x = (sqrt(1+n)-1)/n, which (by a happy
accident, so to speak) is >= 0 and <= 1.

Then argue that the minimum is global...

Sorrow in all lands, and grievous omens.
Great anger in the dragon of the hills,
And silent now the earth's green oracles
That will not speak again of innocence.
David Sutton -- Geomancies

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