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

[egin{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 +\ &frac12left[frac{partial^2 f}{partial x_1^2}Delta x_1^2 + 2frac{partial^2 f}{partial x_1partial x_2}Delta x_1Delta x_2 + frac{partial^2 f}{partial x_2^2}Delta x_2^2 ight] + dots end{aligned} ]

表示成矩阵形式即为

[f(X) = f(X^0) + egin{pmatrix}frac{partial f}{partial x_1}&frac{partial f}{partial x_2}end{pmatrix} egin{pmatrix}Delta x_1\Delta x_2end{pmatrix} + frac12egin{pmatrix}Delta x_1&Delta x_2end{pmatrix}egin{pmatrix}frac{partial^2 f}{partial x_1^2} & frac{partial^2 f}{partial x_1partial x_2}\ frac{partial^2 f}{partial x_1partial x_2} & frac{partial^2 f}{partial x_2^2}end{pmatrix}egin{pmatrix}Delta x_1\ Delta x_2end{pmatrix} + dots ]

其中, 记

[G(X^{(0)}) = egin{pmatrix}frac{partial^2 f}{partial x_1^2} & frac{partial^2 f}{partial x_1partial x_2}\ frac{partial^2 f}{partial x_1partial 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. ( abla f(X^{(0)}) = left[frac{partial f}{partial x_1}, frac{partial f}{partial x_2},dots,frac{partial f}{partial x_n} ight]), 为(f(X))在(X^{(0)})处的梯度.
2. (G(X^{(0)}) = egin{bmatrix}frac{partial^2 f}{partial x_1^2} & frac{partial^2 f}{partial x_1partial x_2} & dots & frac{partial^2 f}{partial x_1partial x_n}\ frac{partial^2 f}{partial x_2partial x_1} & frac{partial^2 f}{partial x^2_2} & dots & frac{partial^2 f}{partial x_2partial x_n} \ vdots & vdots && ddots && vdots\ frac{partial^2 f}{partial x_npartial x_1} & frac{partial^2 f}{partial x_npartial 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} ight|_{(a_1,a_2,dots,a_n)} = 0, j = 1,2,dots, n ]

且有

[A = egin{bmatrix}frac{partial^2 f}{partial x_1^2} & frac{partial^2 f}{partial x_1partial x_2} & dots & frac{partial^2 f}{partial x_1partial x_n}\ frac{partial^2 f}{partial x_2partial x_1} & frac{partial^2 f}{partial x^2_2} & dots & frac{partial^2 f}{partial x_2partial x_n} \ vdots & vdots & ddots & vdots\ frac{partial^2 f}{partial x_npartial x_1} & frac{partial^2 f}{partial x_npartial 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.