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

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Timothy Murphy

Posts: 496
Registered: 12/18/07
Re: An optimization problem
Posted: Sep 8, 2013 10:08 AM
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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.


Agreed.
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/ eircom.net
School of Mathematics, Trinity College, Dublin 2, Ireland




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