<s>[INST] A nonlinear optimization problem with difficult Kuhn-Tucker system of equations I know about the sufficient optimality theorem Kuhn-Tucker, and this problem can use the Kuhn-Tucker theorem directly, but ridiculously, I got stuck on the system of equations to find one root for optimization value. Hope someone can help me or give me some hint to solve it. Thanks.

Find the minimum of $\theta(x,y,z) = (x + 3)^2 + (y-2)^2 +(z-4)^2 + 12$, with $x, y,z \in R$ which satisfies $g(x,y,z) = x^2 + y^2 + z^2 -3x + 2y -8z + 2 \le 0$ and $h(x,y,z) = x  + 3y + 5z = 1$

Using Kuhn-Tucker theorem, I need to solve the system of equation and in equation:
$$
\left\{ 
\begin{array}{c}
\nabla \theta(x) + u\nabla g(x) + vB= 0 \\
g(x) \le 0 \\
Bx = d \\
ug(x) = 0 \\
u \ge 0
\end{array}
\right. 
$$
Here $B = \left[ \begin{matrix}1& 3 &5\end{matrix} \right]$
Which is equivalent to:
$$
\left\{ 
\begin{array}{c}
(2x + 6) + u(2x - 3) + v = 0 (1)\\
(2y - 4) + u(2y + 2) + 3v = 0 (2)\\
(2z - 8) + u(2z - 8) + 5v = 0 (3)\\
x^2 + y^2 + z^2 -3x + 2y -8z + 2 \le 0 (4)\\
u(x^2 + y^2 + z^2 -3x + 2y -8z + 2) = 0 (5)\\
x + 3y + 5z = 1 (6)\\
u \ge 0 (7)
\end{array}
\right. 
$$
I calculated $1 * (1) + 3 * (2) + 5 * (3)$, then use (6), I come to conclusion that $35v - 35u = 44$. From this I can conclude that $u \neq 0$, therefore, we have this system:
$$
\left\{ 
\begin{array}{c}
(2x + 6) + u(2x - 3) + v = 0 \\
(2y - 4) + u(2y + 2) + 3v = 0 \\
(2z - 8) + u(2z - 8) + 5v = 0 \\
x^2 + y^2 + z^2 -3x + 2y -8z + 2 = 0\\
x + 3y + 5z = 1 \\
u \ge 0 
\end{array}
\right. 
$$
And this is where I got stuck
 [/INST] $$\theta_{min}=\theta(x_0,y_0,z_0)=63-\frac{3 \sqrt{132685}}{35}$$
$${x_0}=1-\frac{54 \sqrt{132685}}{7805},\;{y_0}=\frac{61 \sqrt{132685}}{15610}-\frac{5}{2},\;{z_0}=\frac{3}{2}-\frac{3 \sqrt{132685}}{3122}$$
Used free CAS Maxima
http://maxima.sourceforge.net/
</s>