I don't have a solid background about using a KKT theorem. So, I want to check my answer is correct or not.
The problem is the following :
Minimize $f = x+y$ subject to $(x-1)^2 + y^2 \leq 1$, $(x+1)^2 + y^2 \leq 1$, where $(x,y) \in R^2$. Show that the optimal solution $(x,y)$ does not satisfy the conditions of the Karush–Kuhn–Tucker theorem. Explain what hypotheses of the theorem are violated, and why.
My attempt : Let $g_1$ be $(x-1)^2 + y^2 -1$ and $g_2$ be $(x+1)^2 +y^2 -1$. Since there is only one point in this feasible region, namely, $(0,0)$, we can suppose that this point is optimal solution for this convex optimization problem.
By the KKT of gradient form, we already know that $\bigtriangledown f(0, 0)+ \bigtriangledown g_1(0,0) + \bigtriangledown g_2(0,0) = (0,0)$. But, this is impossible in this convex programming problem. Thus, it does not satisfy this condition of KKT gradient form.
Is my attempt correct idea? I'm not sure.