Problem when minimizing user-defined function in Mathematica with Minimize[]

Question

I'm trying to learn how to use the function Minimize in Mathematica.

I first defined the following function in Mathematica:

$$r(x,y,z,i)=\sqrt{(x-X_i)^2+(y-Y_i)^2+(z-Z_i)^2}$$

with the command

r[x_, y_, z_, i_] := Sqrt[(x-Subscript[X, i])^2 + (y-Subscript[Y, i])^2 + (z-Subscript[Z, i])^2]

Then I tried minimizing the following function over $x,y,z$:

$$ f(x,y,z)=(R_i-r(x,y,z,i))^2 $$

with the following command:

Minimize[(Subscript[R, i] - r[x, y, z, 1])^2, {x, y, z}]

The output I was expecting is a circle of points centered around $(X_i,Y_i,Z_i)$ and with radius $R_i$ (since at these points the function attains the value zero), or at least any point on that circle.

However, Mathematica outputs a very long and (for my knowledge level) cryptic solution. I can't think of any reasonable way to post the output here in a human-readable way, so perhaps it's easier if you try doing the commands in your Mathematica and see the output there (if someone thinks it's clearer, I can insert the whole output here, or perhaps a screenshot of it).

So my question is: am I doing something wrong, or is there a way to interpret this output in a meaningful way?

Any help would be appreciated, thank you.

Answer 1

There are two problems here:

1. Your proposed solution is only valid if $R_i > 0$

2. The minimum is attained in every point on a sphere. But Minimize returns only one point, not a set of points. The documentation says,

   Even if the same minimum is achieved at several points, only one is returned.

Simplify the result as Simplify[..., Subscript[R, i] > 0] and you get a clear minimum value: 0. The minimum point looks complicated, but it is valid: it is on the sphere.

If you only want the minimum value, use MinValue.

To get a simpler minimum point, you could provide some constraints which restrict the solution to a single point, e.g.

result = 
  Minimize[{(Subscript[R, i] - r[x, y, z, 1])^2, 
    x == Subscript[X, 1] && y == Subscript[Y, 1]}, {x, y, z}];

Simplify[result, Subscript[R, i] > 0]
(* {0, {x -> Subscript[X, 1], y -> Subscript[Y, 1], 
  z -> -Subscript[R, i] + Subscript[Z, 1]}} *)