黑塞矩陣

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

黑塞矩陣(Hessian Matrix)

黑塞矩陣是一個多元函數的二階偏導數構成的方陣, 描述了函數的局部曲率.

黑塞矩陣常用語牛頓法解決優化問題, 利用黑塞矩陣可判定多元函數的極值問題. 在實際工程問題的優化設計中, 所列的目標函數往往很復雜, 為了使問題簡化, 常常將目標函數在某點鄰域展開成泰勒多項式來逼近原函數, 此時函數在某點泰勒展開式的矩陣形式中會設計到黑塞矩陣.

二維函數\(f(x_1, x_2)\)在\(X^{(0)}(x_1^{(0)}, x_2^{(0)})\)處的泰勒展開式為

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

表示成矩陣形式即為

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

其中, 記

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

\(G(X^{(0)})\)即為\(f(x_1,x_2)\)在\(X^{(0)}\)處的黑塞矩陣.

將結論擴展到多元函數:

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

利用黑塞矩陣判斷多元函數的極值

當多元函數\(f(x_1, x_2, \dots, x_n)\)在點\(M_0(a_1, a_2, \dots, a_n)\)的鄰域內存在連續二階偏導數且滿足:

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

且有

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

則有

1. 當A為正定矩陣時, f在\(M_0\)為極小值;
2. 當A為負定矩陣時, f在\(M_0\)存在極大值;
3. 當A為不定矩陣時, \(M_0\)不是極值點.
4. 當A為半正定矩陣或半負定矩陣時, \(M_0\)是"可疑"極值點.