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

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 ]

Posts: 11,740
Registered: 7/15/05
Re: An optimization problem
Posted: Sep 8, 2013 6:40 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Peter Percival wrote:
>quasi wrote:
>> Peter Percival wrote:
>>> 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.

>>> I know nothing about such problems, but is it not clear that
>>> _if_ there is a minimum,

>> Let X denote the standard unit cube in R^n.
>> Since f is a continuous real-valued function with domain X, and
>> since X is compact, it follows that f achieves a minimum value
>> on X.

>>> it occurs at some point where all the x_i's are equal?
>> No, that's not automatic -- it requires proof.

>Indeed so. What I meant when I said it was clear is that the
>proof is very simple and can be be done in ones head.
>For simplicity's sake let n = 2, x_1 = y and x_2 = z, then my
>claim is that if
> (1 + y^2 + z^2)/(1 + y + z)
>has a minimum, then y = z there. Why is that clear? Because
>if it weren't so then
> (1 + y^2 + z^2)/(1 + y + z) =/= (1 + z^2 + y^2)/(1 + z + y).

No, your appeal to symmetry doesn't support your claim unless
you know that there is a _unique_ point in R^2 for which the
function achieves a minimum.

If all you know is that the function f is symmetric in the
variables y and z and that f achieves a global minimum value
somewhere in the domain, that's not enough to conclude that
the global minimum value can be achieved at some point for
which y = z.


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

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2015. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.