黑塞矩阵

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author: lunar
date: Wed 02 Sep 2020 10:52:12 AM CST

黑塞矩阵(Hessian Matrix)

黑塞矩阵是一个多元函数的二阶偏导数构成的方阵, 描述了函数的局部曲率.

黑塞矩阵常用语牛顿法解决优化问题, 利用黑塞矩阵可判定多元函数的极值问题. 在实际工程问题的优化设计中, 所列的目标函数往往很复杂, 为了使问题简化, 常常将目标函数在某点邻域展开成泰勒多项式来逼近原函数, 此时函数在某点泰勒展开式的矩阵形式中会设计到黑塞矩阵.

二维函数\(f(x_1, x_2)\)在\(X^{(0)}(x_1^{(0)}, x_2^{(0)})\)处的泰勒展开式为

\[\begin{aligned} f(x_1, x_2) = &f(x_1^{(0)}, x_2^{(0)}) + \frac{\partial f}{\partial x_1}\Delta x_1 + \frac{\partial f}{\partial x_2}\Delta x_2 +\\ &\frac12\left[\frac{\partial^2 f}{\partial x_1^2}\Delta x_1^2 + 2\frac{\partial^2 f}{\partial x_1\partial x_2}\Delta x_1\Delta x_2 + \frac{\partial^2 f}{\partial x_2^2}\Delta x_2^2 \right] + \dots \end{aligned} \]

表示成矩阵形式即为

\[f(X) = f(X^0) + \begin{pmatrix}\frac{\partial f}{\partial x_1}&\frac{\partial f}{\partial x_2}\end{pmatrix} \begin{pmatrix}\Delta x_1\\\Delta x_2\end{pmatrix} + \frac12\begin{pmatrix}\Delta x_1&\Delta x_2\end{pmatrix}\begin{pmatrix}\frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\partial x_2}\\ \frac{\partial^2 f}{\partial x_1\partial x_2} & \frac{\partial^2 f}{\partial x_2^2}\end{pmatrix}\begin{pmatrix}\Delta x_1\\ \Delta x_2\end{pmatrix} + \dots \]

其中, 记

\[G(X^{(0)}) = \begin{pmatrix}\frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\partial x_2}\\ \frac{\partial^2 f}{\partial x_1\partial x_2} & \frac{\partial^2 f}{\partial x_2^2}\end{pmatrix} \]

\(G(X^{(0)})\)即为\(f(x_1,x_2)\)在\(X^{(0)}\)处的黑塞矩阵.

将结论扩展到多元函数:

1. \(\nabla f(X^{(0)}) = \left[\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_n}\right]\), 为\(f(X)\)在\(X^{(0)}\)处的梯度.
2. \(G(X^{(0)}) = \begin{bmatrix}\frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\partial x_2} & \dots & \frac{\partial^2 f}{\partial x_1\partial x_n}\\ \frac{\partial^2 f}{\partial x_2\partial x_1} & \frac{\partial^2 f}{\partial x^2_2} & \dots & \frac{\partial^2 f}{\partial x_2\partial x_n} \\ \vdots & \vdots && \ddots && \vdots\\ \frac{\partial^2 f}{\partial x_n\partial x_1} & \frac{\partial^2 f}{\partial x_n\partial x_2} & \dots & \frac{\partial^2 f}{\partial x_n^2}\end{bmatrix}_{X^{(0)}}\) 为函数\(f(X)\)在\(X^{(0)}\)处的黑塞矩阵.

利用黑塞矩阵判断多元函数的极值

当多元函数\(f(x_1, x_2, \dots, x_n)\)在点\(M_0(a_1, a_2, \dots, a_n)\)的邻域内存在连续二阶偏导数且满足:

\[\left.\frac{\partial f}{\partial x_j}\right|_{(a_1,a_2,\dots,a_n)} = 0, j = 1,2,\dots, n \]

且有

\[A = \begin{bmatrix}\frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\partial x_2} & \dots & \frac{\partial^2 f}{\partial x_1\partial x_n}\\ \frac{\partial^2 f}{\partial x_2\partial x_1} & \frac{\partial^2 f}{\partial x^2_2} & \dots & \frac{\partial^2 f}{\partial x_2\partial x_n} \\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial^2 f}{\partial x_n\partial x_1} & \frac{\partial^2 f}{\partial x_n\partial x_2} & \dots & \frac{\partial^2 f}{\partial x_n^2}\end{bmatrix}_{X^{(0)}} \]

则有

1. 当A为正定矩阵时, f在\(M_0\)为极小值;
2. 当A为负定矩阵时, f在\(M_0\)存在极大值;
3. 当A为不定矩阵时, \(M_0\)不是极值点.
4. 当A为半正定矩阵或半负定矩阵时, \(M_0\)是"可疑"极值点.