Explain what happens if you use KKT to solve this problem:
\begin{equation*} \begin{aligned} & \underset{(x,y)\epsilon \mathbb{R}^2}{\text{minimize}} & & x+y \\ & \text{subject to} & & x^2-y+1 \leqslant 0 \\ &&& y-2x \leqslant 0 \end{aligned} \end{equation*}
EDIT: The only issue I had was when I tried to solve this and got y=2x for for slack variables = (0,1/2) and (x,y) = (-1/2,1) for slack variables = (1,0). And since this is a convex program, it should have a single minimum point.