Solve KKT conditions of the following problem

Question

I'm having problems solving the following

$\min f(X) = −3x^2 +y^2 +2z^2 +2(x+y+z) $

subject to $c(X):=x^2+y^2+z^2−1=0$

Now, I get the KKT:

$-6x +2 -2\lambda x = 0 $

$2y +2 -2\lambda y = 0 $

$4z +2 -2\lambda z = 0 $

together with the constraint $c(X):=x^2+y^2+z^2−1=0$.

The problem is that I cannot find a suitable point, (or pair $x, \lambda$) which solve these conditions.

Any ideas ? The main problem I think it is that I have a quadratic constraint that makes everything harder

Answer 1

It may be hard to find the values for $x,y,z,$ and $\lambda$ analytically. To my best knowledge, this is generally not possible for optimization problems with quadratic cost functions with a quadratic constraints.

Matlab's fmincon function gives me $x = -0.9660, y = -0.1986, z = -0.1657,$ and $\lambda = 4.0352.$

Answer 2

Here is a plot of the function from different angles. It seems like it has a unique global minimum and a unique global maximum, consistent with the Lagrangian having two stationary points: