For minimization problem
$minimize$ $x_1+x_2$ with constraint $x_1^2+x_2^2-2=0$
How does it differ from
$minimize$ $x_1+x_2$ with constraint $-x_1^2-x_2^2+2 \geq 0$ ?
For the equality constraint, I got the Lagrangian
$\mathcal{L}(x_1,x_2,\lambda)=x_1+x_2+\lambda(x_1^2+x_2-2)$
and the KKT conditions
$1+2\lambda x_1=0$
$1+2\lambda x_2=0$
$x_1^2+x_2^2-2=0$
So I got possible points are $(1,1),(-1,-1)$, and indeed $(-1,-1)$ is the minimizer.
Is there any difference with $\geq$ inequality?