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?


4

我檢查了 $ \ begin {bmatrix} -2&0 \\ 0&0 \ end {bmatrix} $ 的黑森州,它是負半定的,但根據功能是凸的。我想念什麼?

10

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.