How to Solve a Mixed Kuhn-Tucker/Lagrange Problem with an Inequality Constraint?

[ra_forever8]

Queston From Mathematical Programming:?

Find the minimum value of P(x,y,z)=Z+1/2(x^2+ y^2+1/10 z^2)
constrained by and where r is a positive constant.
Solve the problem when r=1, r=5.
Can you find the solution for general r?
(Note this is a mixed Kuhn-Tucker/ Lagrange problem. The inequality constraints are not trivial and must be taken into account.)

 

[Aaron Bex]

The solution for the general r can be found using the Kuhn-Tucker conditions and Lagrange multipliers.
First, we define the Lagrangian L(x,y,z,λ1,λ2) as follows:

L(x,y,z,λ1,λ2) = P(x,y,z) + λ1(r^2 - x^2 - y^2 - z^2/10) + λ2(x + y + z - 1)

Next, we set the partial derivatives of the Lagrangian with respect to x, y, z, λ1, and λ2 equal to zero and solve for x, y, z, λ1, and λ2:

∂L/∂x = 2x + λ1(-2x) + λ2 = 0
=> x = -λ1/3

∂L/∂y = 2y + λ1(-2y) + λ2 = 0
=> y = -λ1/3

∂L/∂z = 0.2z + λ1(-0.2z) + λ2 = 0
=> z = 5λ1/3

∂L/∂λ1 = r^2 - x^2 - y^2 - z^2