Consider the problem $$\min x^2+x+y+\cdots$$ $$s.t.\ x+y = 0$$ $$x^2-4=0$$
Write the KKT conditions and find the optimal solution.
Attempt
Suppose I did write the KKT conditions.
For the optimal solution, I draw the region and I observed that it's not convex, it's a big N. So I cannot apply the following result.
Let $f:\mathbb R^n\to \mathbb R$ convex and differentiable. Consider the problem to minimize $f(x)$ subject to $x\in X.$ And let X the a convex region, $\overline x\in X$ feasible. Then $\overline x$ is optimal solution if and only if $\nabla f(\overline x)^t(x-\overline x)\ge0$
My question is what can I do in this case?
Thanks for your help
Note:
This was an exam question, and I don't exactly recall the objective function nor if it had this conditions $x,y\ge0$ :( but clearly remember that the region was the N