求$ \min_x (x_1-1)^2+2(x_1^2-x_2)^2 $的Hesse矩阵

首先，计算一阶偏导数：

$$
\begin{aligned}
\frac{\partial}{\partial x_1}[(x_1 - 1)^2 + 2(x_1^2-x_2)^2] &= 2(x_1-1) + 8x_1(x_1^2 - x_2) \\
\frac{\partial}{\partial x_2}[(x_1 - 1)^2 + 2(x_1^2-x_2)^2] &= -8(x_1^2 - x_2)
\end{aligned}
$$

然后，计算二阶偏导数：

$$
\begin{aligned}
&\frac{\partial^2}{\partial x_1^2}[(x_1 - 1)^2 + 2(x_1^2-x_2)^2] = 2 + 8(3x_1^2 - x_2) \\
&\frac{\partial^2}{\partial x_1\partial x_2}[(x_1 - 1)^2 + 2(x_1^2-x_2)^2] = -16x_1 \\
&\frac{\partial^2}{\partial x_2\partial x_1}[(x_1 - 1)^2 + 2(x_1^2-x_2)^2] = -16x_1 \\
&\frac{\partial^2}{\partial x_2^2}[(x_1 - 1)^2 + 2(x_1^2-x_2)^2] = 32
\end{aligned}
$$

因此，该函数的 Hesse 矩阵为：

$$
H = \begin{bmatrix}
2+8(3x_1^2-x_2) & -16x_1 \\
-16x_1 & 32
\end{bmatrix}
$$