Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

Question

I want to use Minimize where the variables to minimize are indices pointing into an array.

Here a MWE that hopefully shows what my problem is.

vars = u@# & /@ Range[3];
cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 
                        3 - 1}], 
                  1 <= # <= 3 & /@ vars };
vec1 = {1, 2, 3}; vec2 = {1, 2, 3};
Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), 
          cons}, vars, Integers]

The error I get:

Part::pkspec1: The expression u[1] cannot be used as a part specification. >>

Answer 1

Ok,

it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]].

The working MWE would then look like the following:

vars = u@# & /@ Range[3];
cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 
                       3 - 1}], 
                 1 <= # <= 3 & /@ vars};
vec1 = {1, 2, 3}; vec2 = {1, 2, 3};
NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ 
           Range[1, 3]), cons}, vars, Integers ]

which yields correctly:

{0., {u[1] -> 1, u[2] -> 2, u[3] -> 3}}

See also an answer to this post by Mr.Wizard:

Prevent Part[] from trying to extract parts of symbolic expressions