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

*To*: mathgroup at smc.vnet.net*Subject*: [mg96784] Re: [mg96777] Re: no message from Minimize[] on a weird function(x^x)*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Wed, 25 Feb 2009 04:01:01 -0500 (EST)*References*: <gnr1d2$ep0$1@smc.vnet.net> <200902241048.FAA23747@smc.vnet.net>

ADL wrote: > Beyond what others have already commented, I would say that the > problem stems with the definition of 0^0 which, in Mathematica, is > Indeterminate. I do not think the tests below show that to be the issue. > 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)}} 1/E<1, so there is no reason to expect that min to be at the origin. > Minimize[{Abs[x^x], x < 0}, x] // InputForm > {0, {x -> -Infinity}} 0<1, so... > Minimize[{Abs[x^x], x >= 0}, x] // InputForm > Minimize[{Abs[x^x], x >= 0}, x] These simply show that Minimize, at heart an exact algebraic function, does not know how to handle the input. > 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 Nothing to do with 0^0 or holes in the domain. Try a variation of this function but with extremum, in a limiting sense, at 0, In[10]:= Maximize[{x^x, 0 < x <= 1/2}, x] During evaluation of In[10]:= Maximize::wksol: Warning: There is no maximum in the region in which the objective function is defined and the constraints are satisfied; returning a result on the boundary. >> Out[10]= {1, {x -> 0}} When x<0 the function is not in general real valued, hence problems arise there. NMinimize, howeverm can do sensible things with variants. Here are two such. In[16]:= NMinimize[{Re[x^x] + Im[x^x], -1 <= x <= 0}, x, Method -> "DifferentialEvolution"] Out[16]= {-1.79458, {x -> -0.687465}} In[17]:= NMinimize[{Re[x^x] - Im[x^x], -1 <= x <= 0}, x, Method -> "DifferentialEvolution"] Out[17]= {-1., {x -> -1.}} Daniel Lichtblau Wolfram Research

**References**:**Re: no message from Minimize[] on a weird function(x^x) !?!***From:*ADL <alberto.dilullo@tiscali.it>