The Math Forum

Search All of the Math Forum:

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

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

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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 ]
Timothy Murphy

Posts: 657
Registered: 12/18/07
Re: An optimization problem
Posted: Sep 8, 2013 10:08 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

quasi wrote:

> 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.

But in this case one can see that any minimum in the _interior_
of the domain must have x_1 = ... = x_n by partial differentiation.
Let u = 1 + sum x_i, v = 1 + sum x_i^2
Then f = v/u and so log f = log v - log u.
Thus df/dx_i = 2x_i/v - 1/u and so at a stationary point
x_i = v/2u for all i.
So at a stationary point either the x_i are equal,
or one of them is 0 or 1.
In the latter case one can still apply the argument to see that
all the x_i not equal to 1 or 0 are equal.
So the problem reduces to finding the minimum of (r + sx^2)/(r + sx),
which is not difficult.

Timothy Murphy
e-mail: gayleard /at/
School of Mathematics, Trinity College, Dublin 2, Ireland

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

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.