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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96743] Re: [mg96724] no message from Minimize[] on a weird function(x^x)
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Mon, 23 Feb 2009 05:03:51 -0500 (EST)
  • Reply-to: hanlonr at cox.net

f[x_] := x^x

Minimize[f[x], x]

Minimize[x^x,x]


Minimize returns unevaluated because it is trying to provide a symbolic result and cannot solve the problem. You need to simplify the problem by providing the appropriate constraint.

Minimize[{f[x], x > 0}, x]

{E^(-E^(-1)), {x -> 1/E}}

% // N

{0.692201,{x->0.367879}}


As with Minimize, MinValue and ArgMin also need the constraint

{MinValue[f[x], x], ArgMin[f[x], x]}

{MinValue[x^x,x],ArgMin[x^x,x]}


{MinValue[{f[x], x > 0}, x], ArgMin[{f[x], x > 0}, x]}

{E^(-E^(-1)), 1/E}


Numerical techniques do not need the constraint presumably because their search starts with x > 0

FindMinimum[f[x], x]

{0.692201,{x->0.367879}}

FindArgMin[f[x], x]

{0.367879}

FindMinValue[f[x], x]

0.692201

NMinimize[f[x], x]

{0.692201,{x->0.367879}}

NMinValue[f[x], x]

0.692201


Bob Hanlon

---- congruentialuminaire 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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