多元函数的极值问题: 通常极值与条件极值
通常极值
定理 设函数 \(f(\boldsymbol{x})=f(x_1,\cdots,x_n)\) 在区域 \(D\subset\mathbb{R}^n\) 内具有二阶连续偏导数, 且 \(f'(\boldsymbol{x}_0)\) \((\boldsymbol{x}_0\in D)\), 再设 \(f(\boldsymbol{x})\) 在 \(\boldsymbol{x}_0\) 处的Hessi矩阵满秩, 则
(1) 当 \(\boldsymbol{H}_f(\boldsymbol{x}_0)\) 正定时, \(f(\boldsymbol{x})\) 在 \(\boldsymbol{x}_0\) 取极小值;
(2) 当 \(\boldsymbol{H}_f(\boldsymbol{x}_0)\) 负定时, \(f(\boldsymbol{x})\) 在 \(\boldsymbol{x}_0\) 取极大值;
(3) 当 \(\boldsymbol{H}_f(\boldsymbol{x}_0)\) 不定时, \(f(\boldsymbol{x})\) 在 \(\boldsymbol{x}_0\) 不是极值;
- Hessi矩阵定义如下: \[\boldsymbol{H}_f(\boldsymbol{x}_0)=\begin{bmatrix}\frac{\partial f(\boldsymbol{x}_0)}{\partial x_1^2} & \frac{\partial f(\boldsymbol{x}_0)}{\partial x_1\partial x_2} & \cdots & \frac{\partial f(\boldsymbol{x}_0)}{\partial x_1\partial x_n}\\ \frac{\partial f(\boldsymbol{x}_0)}{\partial x_2\partial x_1} & \frac{\partial f(\boldsymbol{x}_0)}{\partial x_2^2} & \cdots & \frac{\partial f(\boldsymbol{x}_0)}{\partial x_2\partial x_n}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial f(\boldsymbol{x}_0)}{\partial x_n\partial x_1} & \frac{\partial f(\boldsymbol{x}_0)}{\partial x_n\partial x_1} & \cdots & \frac{\partial f(\boldsymbol{x}_0)}{\partial x_n^2}\\ \end{bmatrix}.\]
条件极值
定理 设函数 \(f(\boldsymbol{x})\), \(\boldsymbol{\varphi}(\boldsymbol{x})=(\varphi_1(\boldsymbol{x}),\cdots,\varphi_m(\boldsymbol{x}))\) 在区域 \(D\subset\mathbb{R}^n~(m<n)\) 内具有各个连续偏导数, 再设 \(\boldsymbol{x}_0=(x_1^0,\cdots,x_n^0)\in D\) 为 \(f(\boldsymbol{x})\) 在约束条件 \[\begin{cases}\varphi_1(\boldsymbol{x})=0,\\ \varphi_2(\boldsymbol{x})=0,\\ \cdots\cdots\\ \varphi_m(\boldsymbol{x})=0.\end{cases}\] 下的极值点, 且 \(\boldsymbol{\varphi}'(\boldsymbol{x}_0)\) 的秩为 \(m\), 则存在常数 \(\lambda_1,\cdots,\lambda_m\in\mathbb{R}\), s.t. 在 \(\boldsymbol{x}_0\) 处有如下等式成立: \[\begin{cases} \frac{\partial f(\boldsymbol{x}_0)}{\partial x_i}+\sum\limits_{j=1}^m \lambda_j\frac{\partial \varphi_j(\boldsymbol{x}_0)}{\partial x_i}=0, & i=1,2,\cdots,n, \\ \varphi_j(\boldsymbol{x}_0)=0, & j=1,2,\cdots,m. \end{cases}\]
Lagrange乘数法
构造函数 \[F(x_1,\cdots,x_n,\lambda_1,\cdots,\lambda_m)=f(\boldsymbol{x})+\sum_{j=1}^m\lambda_j\varphi_j(\boldsymbol{x}),\]
则 \(\boldsymbol{x}_0\) 是极值点的必要条件如下: \[\begin{cases}\frac{\partial F(\boldsymbol{x}_0)}{\partial x_i}=0 & (i=1,2,\cdots,n),\\ \frac{\partial F(\boldsymbol{x}_0)}{\partial\lambda_j}=0 & (j=1,2,\cdots,m).\\ \end{cases}\]