# How to check for convexity of the inequality constraint $−x^2+y−1\ge0$ for a minimization objective function?

I checked the Hessian which is $$\begin{bmatrix}-2&0\\0&0\end{bmatrix}$$ which is negative semidefinite but according to the sketch of the function it is convex. What am I missing?

• The function on the LHS is concave so the set specified by the inequality is convex. – ErlingMOSEK Apr 7 '20 at 12:20

• $$\{x:g(x) \le 0\}$$ is convex if $$g$$ is convex.
\begin{align}\{(x,y): -x^2+y-1 \ge 0\}&=\{(x,y): -(-x^2+y-1 ) \le 0\} \\ &=\{(x,y): x^2-y+1 \le 0\} \end{align}
If you compute the Hessian of $$x^2-y+1$$, you will obtain $$\begin{bmatrix} 2 & 0 \\ 0 & 0\end{bmatrix} \succeq 0$$, hence the corresponding region is a convex set.