Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Peter Percival

Posts: 1,219
Registered: 10/25/10
Re: An optimization problem
Posted: Sep 8, 2013 5:13 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

quasi wrote:
> William Elliot wrote:
>> 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?

>>
>> 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



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.