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