Maximize and minimize R6 function using Lagrange Multipliers

Question

I've been trying to maximize and minimize the following function using Lagrange Multipliers, in Wolfram Mathematica 11.3:

d = Sqrt[x^2 + y^2 + z^2 + u^2 + v^2 + w^2]

Subject to the following constraints:

g = x + y + z + u + v + w - 1 and h = x^2 + y^2 + z^2 + u^2 + v^2 - 1

What I've tried so far is the following:

Gd = {D[d, x], D[d, y], D[d, z], D[d, u], D[d, v], D[d, w]}

Gg = {D[g, x], D[g, y], D[g, z], D[g, u], D[g, v], D[g, w]}


Gh = {D[h, x], D[h, y], D[h, z], D[h, u], D[h, v]}

system = {Gd[[1]] == \[Lambda] Gg[[1]] + \[Micro] Gh[[1]], 
   Gd[[2]] == \[Lambda] Gg[[2]] + \[Micro] Gh[[2]], 
   Gd[[3]] == \[Lambda] Gg[[3]] + \[Micro] Gh[[3]], 
   Gd[[4]] == \[Lambda] Gg[[4]] + \[Micro] Gh[[4]], 
   Gd[[5]] == \[Lambda] Gg[[5]] + \[Micro] Gh[[5]], 
   Gd[[6]] == \[Lambda] Gg[[6]], g == 0, h == 0};
% // MatrixForm

And then a little Solve function:

sist = Solve[system, {x, y, z, u, v, w, \[Lambda], \[Micro]}]

After executing, it just keeps running forever. Any ideas on how to make it easier?

Answer 1

For the minimum, you can use NMinValue. I changed it up to use some Region functionality:

NMinValue[{Sqrt[v.v+w^2], Total[v]+w==1, v ∈ Sphere[5]},{v, w}]

1.

Or, you can use explicit scalars:

NMinValue[{Sqrt[x^2+y^2+z^2+u^2+v^2+w^2], x+y+u+v+w==1&&x^2+y^2+z^2+u^2+v^2==1},{x,y,z,u,v,w}]

1.

I'm not sure if using MinValue would ever finish.

For the maximum, you can use MaxValue:

MaxValue[{Sqrt[v.v+w^2], Total[v]+w==1, v ∈ Sphere[5]},{v, w}]

Sqrt[7 + 2 Sqrt[5]]

Or, using explicit scalars:

MaxValue[{Sqrt[x^2+y^2+z^2+u^2+v^2+w^2], x+y+z+u+v+w==1&&x^2+y^2+z^2+u^2+v^2==1},{x,y,z,u,v,w}]

Sqrt[7 + 2 Sqrt[5]]

Use NMinimize and Maximize if you want the location of the extrema as well as the value.