# 如何為最小化目標函數檢查不等式約束$-x ^ 2 + y-1 \ ge0$的凸度？

I think you are trying to use the following property:

• $$\{x:g(x) \le 0\}$$ is convex if $$g$$ is convex.

Note the direction of the inequality.

Notice that

\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.