The following non-linear optimalisation problem:
\begin{aligned} & \quad \min x_1 \\ & \text { } x_1+x_2=0 \\ & \quad\left(x_1^2+x_2^2-4\right)^2=0 \end{aligned}
The task is to find the KKT-conditions.
The solution says that:
\begin{aligned} & {\left[\begin{array}{l} 1 \\ 0 \end{array}\right]=\lambda_1\left[\begin{array}{l} 1 \\ 1 \end{array}\right]+2 \lambda_2\left(x_1^2+x_2^2-4\right)\left[\begin{array}{l} 2 x_1 \\ 2 x_2 \end{array}\right]} \\ & x_1+x_2=1 \\ & \left(x_1^2+x_2^2-4\right)^2=0 . \end{aligned}
I see where the first and last equations come from, but I do not see yet how the second one is derived.
Question: In the problem describtion it is given that $x_1+x_2 = 0$, why is is so that in the KKT-condition, the equations becomes $x_1+x_2 = 1$?
Feedback on this would be helpful!
