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Re: no message from Minimize[] on a weird function(x^x) !?!

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96777] Re: no message from Minimize[] on a weird function(x^x) !?!
  • From: ADL <alberto.dilullo at tiscali.it>
  • Date: Tue, 24 Feb 2009 05:48:31 -0500 (EST)
  • References: <gnr1d2$ep0$1@smc.vnet.net>

Beyond what others have already commented, I would say that the
problem stems with the definition of 0^0 which, in Mathematica, is
Indeterminate. In fact, even if the value of the function is
constrained to be real positive, Minimize behavior is influenced by
the value at 0:

Minimize[{Abs[x^x], x > 0}, x] // InputForm
{E^(-E^(-1)), {x -> E^(-1)}}

Minimize[{Abs[x^x], x < 0}, x] // InputForm
{0, {x -> -Infinity}}

Minimize[{Abs[x^x], x >= 0}, x] // InputForm
Minimize[{Abs[x^x], x >= 0}, x]

In fact,

0^0
Indeterminate

but
Limit[Abs[x^x], x -> 0]
1.

In conclusion, Minimize does not appear to compute limits and cannot
deal with "holes" in the function's domain.

ADL


On 22 Feb, 09:12, congruentialumina... at yahoo.com wrote:
> Hello MathGroup:
>
> I have:
>
> f[x_]=x^x
> Plot[f[x],{x,-3,3.}]
>
> What makes this a weird function is that when x<0, the function is
> discontinuous and non-differentiable and has a global minimum at -1.
>
> To answer the question: "what is the minimum of this function", I
> tried
>
> FindMinimum[f[x],{x,2}] (* this appears correct *)> {0.692201, {x -> 0.367879}}
>
> FindMinimum[f[x],{x,2}] (* this complains about the gradient, but
> appears correct *)> FindMinimum::nrgnum: The gradient is not a vector of real numbers at {x} = {-1.}. >>
> > {-1., {x -> -1.}}
>
> NMinimize[f[x], x] (* this gives the minimum in the positive domain *)> {0.692201, {x -> 0.367879}}
>
> Minimize[f[x], x] (* this gives no answer and no error message *)
>
> > Minimize[x^x, x]
>
> Is this expected behavior?
>
> TIA.
>
> Roger Williams
> Franklin Laboratory



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