
Re: An optimization problem
Posted:
Sep 8, 2013 10:08 AM


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 email: gayleard /at/ eircom.net School of Mathematics, Trinity College, Dublin 2, Ireland

