黑塞矩阵

海森矩阵（德语：Hesse-Matrix；英语：Hessian matrix 或 Hessian），又译作黑塞矩阵、海塞（赛）矩阵或海瑟矩阵等，是一个由多变量实值函数的所有二阶偏导数组成的方阵，由德国数学家奥托·黑塞引入并以其命名。

定义

假设有一实值函数$f(x_{1},x_{2},\dots ,x_{n})\,$，如果 $f\,$的所有二阶偏导数都存在并在定义域内连续，那么函数$f\,$的黑塞矩阵为

$$\mathbf {H} ={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}\,$$

或使用下标记号表示为

$$\mathbf {H} _{ij}={\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}$$

显然黑塞矩阵 $\mathbf {H} \,$是一个$n\times n\,$方阵。黑塞矩阵的行列式被称为黑塞式（英语：Hessian），而需注意的是英语环境下使用Hessian一词时可能指上述矩阵也可能指上述矩阵的行列式。

性质

由高等数学知识可知，若一元函数$f(x)\,$在$x=x_{0}\,$点的某个邻域内具有任意阶导数，则函数$f(x)\,$在$x=x_{0}\,$点处的泰勒展开式为

$$f(x)=f(x_{0})+f'(x_{0})\Delta x+{\frac {f''(x_{0})}{2!}}\Delta x^{2}+\cdots \,$$

其中，$\Delta x=x-x_{0}\,$。

同理，二元函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点处的泰勒展开式为

$$f(x_{1},x_{2})=f(x_{10},x_{20})+f_{x_{1}}(x_{10},x_{20})\Delta x_{1}+f_{x_{2}}(x_{10},x_{20})\Delta x_{2}+{\frac {1}{2}}[f_{x_{1}x_{1}}(x_{10},x_{20})\Delta x_{1}^{2}+2f_{x_{1}x_{2}}(x_{10},x_{20})\Delta x_{1}\Delta x_{2}+f_{x_{2}x_{2}}(x_{10},x_{20})\Delta x_{2}^{2}]+\cdots \,$$

其中，$\Delta x_{1}=x_{1}-x_{10}\,$，$\Delta x_{2}=x_{2}-x_{20}\,$，$f_{x_{1}}={\frac {\partial f}{\partial x_{1}}}\,$，$f_{x_{2}}={\frac {\partial f}{\partial x_{2}}}\,$，$f_{x_{1}x_{1}}={\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\,$，$f_{x_{2}x_{2}}={\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\,$，$f_{x_{1}x_{2}}={\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}}={\frac {\partial ^{2}f}{\partial x_{2}\partial x_{1}}}\,$。

将上述展开式写成矩阵形式，则有

$$f(x)=f(x_{0})+\nabla f(x_{0})^{\mathrm {T} }\Delta x+{\frac {1}{2}}\Delta x^{\mathrm {T} }G(x_{0})\Delta x+\cdots$$

其中，$\Delta x={\begin{bmatrix}\Delta x_{1}\\\\\Delta x_{2}\end{bmatrix}}\,$，$\Delta x^{\mathrm {T} }={\begin{bmatrix}\Delta x_{1}&\Delta x_{2}\end{bmatrix}}\,$是$\Delta x$的转置，$\nabla f(x_{0})={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}\\\\{\frac {\partial f}{\partial x_{2}}}\end{bmatrix}}\,$是函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$的梯度，矩阵

$$G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{bmatrix}}_{x_{0}}\,$$

即函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点处的$2\times 2\,$黑塞矩阵。它是由函数$f(x_{1},x_{2})$在$x_{0}(x_{10},x_{20})$点处的所有二阶偏导数所组成的方阵。

由函数的二次连续性，有

$${\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}}={\frac {\partial ^{2}f}{\partial x_{2}\partial x_{1}}}$$

所以，黑塞矩阵$G(x_{0})\,$为对称矩阵。

将二元函数的泰勒展开式推广到多元函数，函数$f(x_{1},x_{2},\cdots ,x_{n})\,$在$x_{0}(x_{1},x_{2},\cdots ,x_{n})\,$点处的泰勒展开式为

$$f(x)=f(x_{0})+\nabla f(x_{0})^{\mathrm {T} }\Delta x+{\frac {1}{2}}\Delta x^{\mathrm {T} }G(x_{0})\Delta x+\cdots \,$$

其中，$$\nabla f(x_{0})={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}&{\frac {\partial f}{\partial x_{2}}}&\cdots &{\frac {\partial f}{\partial x_{n}}}\end{bmatrix}}_{x_{0}}^{T}\,$$ 为函数$f(x)$在$x_{0}(x_{1},x_{2},\cdots ,x_{n})\,$点的梯度，

$$G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}_{x_{0}}\,$$

为函数$f(x)\,$在$x_{0}(x_{1},x_{2},\cdots ,x_{n})\,$点的$n\times n\,$黑塞矩阵。若函数有$n\,$次连续性，则函数的$n\times n\,$黑塞矩阵是对称矩阵。

说明：在优化设计领域中，黑塞矩阵常用$G\,$表示，且梯度有时用$g\,$表示。

函数$f\,$的黑塞矩阵和雅可比矩阵有如下关系：

$$\mathrm {H} (f)=\mathrm {J} (\nabla f)^{T}\,$$

即函数$f\,$的黑塞矩阵等于其梯度的雅可比矩阵。

应用

函数的极值条件

对于一元函数$f(x)\,$，在给定区间内某$x=x_{0}\,$点处可导，并在$x=x_{0}\,$点处取得极值，其必要条件是

$$f'(x_{0})=0\,$$

即函数$f(x)\,$的极值必定在驻点处取得，或者说可导函数$f(x)\,$的极值点必定是驻点；但反过来，函数的驻点不一定是极值点。检验驻点是否为极值点，可以采用二阶导数的正负号来判断。根据函数$f(x)\,$在$x=x_{0}\,$点处的泰勒展开式，考虑到上述极值必要条件，有

$$f(x)=f(x_{0})+{\frac {f''(x_{0})}{2!}}\Delta x^{2}+\cdots \,$$

若$f(x)\,$在$x=x_{0}\,$点处取得极小值，则要求在$x=x_{0}\,$某一邻域内一切点$x\,$都必须满足

$$f(x)-f(x_{0})>0\,$$

即要求

$${\frac {f''(x_{0})}{2!}}\Delta x^{2}>0\,$$

亦即要求

$$f''(x_{0})>0\,$$

$f(x)\,$在$x=x_{0}\,$点处取得极大值的讨论与之类似。于是有极值充分条件：

设一元函数$f(x)\,$在$x=x_{0}\,$点处具有二阶导数，且$f'(x_{0})=0\,$，$f''(x_{0})\neq 0\,$，则

1. 当$f''(x_{0})>0\,$时，函数$f(x)\,$在$x=x_{0}\,$处取得极小值；
2. 当$f''(x_{0})<0\,$时，函数$f(x)\,$在$x=x_{0}\,$处取得极大值。

而当$f''(x_{0})=0\,$时，无法直接判断，还需要逐次检验其更高阶导数的正负号。由此有一个规律：若其开始不为零的导数阶数为偶数，则驻点是极值点；若为奇数，则为拐点，而不是极值点。

对于二元函数$f(x_{1},x_{2})\,$，在给定区域内某$x_{0}(x_{10},x_{20})\,$点处可导，并在$x_{0}(x_{10},x_{20})\,$点处取得极值，其必要条件是

$$f_{x_{1}}(x_{0})=f_{x_{2}}(x_{0})=0\,$$

即

$$\nabla f(x_{0})=0\,$$

同样，这只是必要条件，要进一步判断$x_{0}(x_{10},x_{20})\,$是否为极值点需要找到取得极值的充分条件。根据函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点处的泰勒展开式，考虑到上述极值必要条件，有

$$f(x_{1},x_{2})=f(x_{10},x_{20})+{\frac {1}{2}}[f_{x_{1}x_{1}}(x_{0})\Delta x_{1}^{2}+2f_{x_{1}x_{2}}(x_{0})\Delta x_{1}\Delta x_{2}+f_{x_{2}x_{2}}(x_{0})\Delta x_{2}^{2}]+\cdots \,$$

设$A=f_{x_{1}x_{1}}(x_{0})\,$，$B=f_{x_{1}x_{2}}(x_{0})\,$，$C=f_{x_{2}x_{2}}(x_{0})\,$，则

$$f(x_{1},x_{2})=f(x_{10},x_{20})+{\frac {1}{2}}[A\Delta x_{1}^{2}+2B\Delta x_{1}\Delta x_{2}+C\Delta x_{2}^{2}]+\cdots \,$$

或

$$f(x_{1},x_{2})=f(x_{10},x_{20})+{\frac {1}{2A}}[(A\Delta x_{1}+B\Delta x_{2})^{2}+(AC-B^{2})\Delta x_{2}^{2}]+\cdots \,$$

若$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点处取得极小值，则要求在$x_{0}(x_{10},x_{20})\,$某一邻域内一切点$x\,$都必须满足

$$f(x_{1},x_{2})-f(x_{10},x_{20})>0\,$$

即要求

$${\frac {1}{2A}}[(A\Delta x_{1}+B\Delta x_{2})^{2}+(AC-B^{2})\Delta x_{2}^{2}]>0\,$$

亦即要求$A>0\,$，$AC-B^{2}>0\,$

即
$$\left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}>0\,$$

$${\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}-({\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}})^{2}\end{bmatrix}}_{x_{0}}>0\,$$

此条件反映了$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点处的黑塞矩阵$G(x_{0})\,$的各阶主子式都大于零，即对于

$$G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{bmatrix}}_{x_{0}}\,$$

要求
$$\left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}>0\,$$

$$|G(x_{0})|={\begin{vmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{vmatrix}}_{x_{0}}>0\,$$

$f((x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点处取得极大值的讨论与之类似。于是有极值充分条件：

设二元函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点的邻域内连续且具有一阶和二阶连续偏导数，又有$f_{x_{1}}(x_{0})=f_{x_{2}}(x_{0})=0\,$，同时令$A=f_{x_{1}x_{1}}(x_{0})\,$，$B=f_{x_{1}x_{2}}(x_{0})\,$，$C=f_{x_{2}x_{2}}(x_{0})\,$，则

1. 当$A>0\,$，$AC-B^{2}>0\,$时，函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$处取得极小值；
2. 当$A<0\,$，$AC-B^{2}>0\,$时，函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$处取得极大值。

此外可以判断，当$AC-B^{2}<0\,$时，函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点处没有极值，此点称为鞍点。而当$AC-B^{2}=0\,$时，无法直接判断，对此，补充一个规律：当$AC-B^{2}=0\,$时，如果有$A\equiv 0\,$，那么函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$有极值，且当$C>0\,$有极小值，当$C<0\,$有极大值。

由线性代数的知识可知，若矩阵$G(x_{0})\,$满足
$$\left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}>0\,$$

$${\begin{vmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{vmatrix}}_{x_{0}}>0\,$$

则矩阵$G(x_{0})\,$是正定矩阵，或者说矩阵$G(x_{0})\,$正定。

若矩阵$G(x_{0})\,$满足
$$\left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}<0\,$$

$${\begin{vmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{vmatrix}}_{x_{0}}>0\,$$

则矩阵$G(x_{0})\,$是负定矩阵，或者说矩阵$G(x_{0})\,$负定。

于是，二元函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点处取得极值的条件表述为：二元函数$f(x_{1},x_{2})\,$在$x_{0}(x_{10},x_{20})\,$点处的黑塞矩阵正定，则取得极小值；在$x_{0}(x_{10},x_{20})\,$点处的黑塞矩阵负定，则取得极大值。

对于多元函数$f(x_{1},x_{2},\cdots ,x_{n})\,$，若在$x_{0}(x_{1},x_{2},\cdots ,x_{n})\,$点处取得极值，则极值存在的必要条件为

$$\nabla f(x_{0})={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}&{\frac {\partial f}{\partial x_{2}}}&\cdots &{\frac {\partial f}{\partial x_{n}}}\end{bmatrix}}_{x_{0}}^{T}=0\,$$

取得极小值的充分条件为

$$G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}_{x_{0}}\,$$

正定，即要求$G(x_{0})\,$的各阶主子式都大于零，即
$$\left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}>0\,$$

$${\begin{vmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{vmatrix}}_{x_{0}}>0\,$$

$$\vdots$$

$$|G(x_{0})|>0\,$$
取得极大值的充分条件为

$$G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}_{x_{0}}\,$$

负定。

拓展阅读

• 雅可比矩阵
• 梯度