I am trying to find the minimum of $-x_1$ with restrictions $\bar g\leq\bar 0$ so that
$\bar g=\begin{pmatrix} (x_1+2)^2+(x_2-4)^2-20\\ (x_1+2)^2+x_2^2-20\\ -x_1\end{pmatrix}\leq \begin{pmatrix}0\\0\\0\\\end{pmatrix}=\bar0$
I used KKT conditions to solve this puzzle below so $x_1=\frac{4\mu_1+4\mu_2-1-\mu_3}{2(\mu_1+\mu_2)}$ and $x_2=\frac{4\mu_1}{\mu_1+\mu_2}$ where $\mu_i\in\mathbb R\forall i$. I know from the graphical plot that the solution is something like $(1.5,1.5)$ but I cannot see how I can get such a solution from the equations for $x_1$ and $x_2$.
I followed this part of Wikipedia here, source here, about necessary conditions but I am stack how to find the minimum now. How to find it now with the necessary equations for the optimal point $(x_1,x_2)$?
My calculations
Updates
Wok suggested complementary slackness -assumption $\mu_i g_i(x^*)=0, i=1,2,3$ and dual -feasibility assumption $\mu_i\geq 0,i=1,2,3$. I cannot yet see what it helps here.
I can solve the intersection point just by solving the equations, proof here, but I cannot see how the KKT -way really make a difference in comparison to solving in the easy way, really puzzled!
KKT is some sort of generalization of Lagrangean, example here, trying to understand what is happening...