Why can't I define a function using the Minimization function in Mathematica?

Question

Suppose I define a function in this way:

function[x_,y_]=Minimize[E^(x^2 + y^2 + z^2 + w^2 + x*y + x*z + x*w + y*z + y*w + z*w), {z,w}]

The output in Mathematica will give me the minimal value of the expression and the values for z and w that minimize the expression. If I want to extract just the minimum value for this output, I can call function[1,1][[1]] and this will return the minimal value. Now, suppose I want to define a new function that returns this minimal value, i.e., define:

function2[x_,y_] = function[x,y][[1]]

If I call function2[1,1], I don't get the minimal value that was output when calling function[1,1][[1]], but rather an expression involving the variables z and w. Is there any way to fix this and accomplish my goal of defining a new function that extracts the minimal value from Minimize?

Answer 1

f[x_?NumericQ, y_?NumericQ] := 
 NMinimize[
   E^(x^2 + y^2 + z^2 + w^2 + x*y + x*z + x*w + y*z + y*w + z*w), {z, 
    w}][[1]];
f[1,1]

Answer 2

The problem is that your definitions use Set (=) rather than SetDelayed (:=) in their definitions. = evaluates its right hand side on the spot. You get away with that for function because Minimize can't do the job for unknown x and y, so evaluation does nothing. But in function2, Part extracts the first part of the Minimize expression, and that becomes the definition. It works if you do:

function2[x_, y_] := function[x, y][[1]]

because := delays evaluation until you "call" function2. I'd recommend using := in the definition of function also, because the fact that your definition works is a bit of an accident.