微积分笔记（42）——多变量函数的微分学（5）

Contents

多变量函数的微分学

带约束条件的极值问题

实例

$$S = xy + \frac{2(x + y)V}{xy}$$

条件极值问题

$$\begin{cases} \max f(\mathbf{x}, \mathbf{y}) \\ \pmb{\Phi}(\mathbf{x}, \mathbf{y}) = \mathbf{0} \end{cases} \text{或} \begin{cases} \min f(\mathbf{x}, \mathbf{y}) \\ \pmb{\Phi}(\mathbf{x}, \mathbf{y}) = \mathbf{0} \end{cases} \tag{M}$$

问题分析（$m = 1$）

$$\frac{\partial \Phi}{\partial y}(\mathbf{x}_0, y_0) \not = 0$$

$$\frac{\partial F}{\partial x_i}(\mathbf{x}_0) = \frac{\partial f}{\partial x_i} (\mathbf{x}_0, y_0) + \frac{\partial f}{\partial y}(\mathbf{x}_0, y_0) \frac{\partial y}{\partial x_i} (\mathbf{x}_0) = 0,\ \ i = 1, \cdots, n$$

$$\frac{\partial y}{\partial x_i} (\mathbf{x}_0) = -\frac{\partial \Phi}{\partial x_i} (\mathbf{x}_0, y_0) \Big / \frac{\partial \Phi}{\partial y} (\mathbf{x}_0, y_0)$$

$$\frac{\partial f}{\partial x_i} (\mathbf{x}_0, y_0) – \frac{\partial f}{\partial y}(\mathbf{x}_0, y_0) \frac{\partial \Phi}{\partial x_i} (\mathbf{x}_0, y_0) \Big / \frac{\partial \Phi}{\partial y} (\mathbf{x}_0, y_0) = 0$$

$$\lambda = – \frac{\partial f}{\partial y}(\mathbf{x}_0, y_0) \Big / \frac{\partial \Phi}{\partial y} (\mathbf{x}_0, y_0) = 0$$

$$\frac{\partial f}{\partial x_i} (\mathbf{x}_0, y_0) + \lambda \frac{\partial \Phi}{\partial x_i} (\mathbf{x}_0, y_0) = 0$$

$$J f(\mathbf{x}_0, y_0) + \lambda J \Phi(\mathbf{x}_0, y_0) = 0$$
（必要条件）

定理（条件极值必要定理）

$$J f(\mathbf{x}_0, \mathbf{y}_0) + \Lambda J \pmb{\Phi}(\mathbf{x}_0, \mathbf{y}_0) = \mathbf{0}$$
：令 $\pmb{\Phi} = \begin{pmatrix}\varphi_1 & \cdots & \varphi_m\end{pmatrix}^T, \Lambda = \begin{pmatrix}\lambda_1 & \cdots & \lambda_m\end{pmatrix}$，上式化为：
$$J(f + \lambda_1 \varphi_1 + \cdots + \lambda_m \varphi_m) = \mathbf{0}$$
（在 $P$ 点）

Lagrange 乘数法

Lagrange 数乘法（引入辅助函数）

$$L(\mathbf{z}, \Lambda) = f(\mathbf{z}) + \Lambda \pmb{\Phi}(\mathbf{z}), (\mathbf{z}, \Lambda) \in D \times \mathbb{R}^m$$

$$J_\mathbf{z} L(\mathbf{z}_0, \Lambda) = J_\mathbf{z} f(\mathbf{z}_0) + \Lambda \Phi(\mathbf{z}_0) = \mathbf{0}$$

$$J_\Lambda L(\mathbf{z}_0, \Lambda) = \pmb{\Phi}(\mathbf{z}_0) = \mathbf{0}$$

$$J L(\mathbf{z}_0, \Lambda) = \mathbf{0}$$
——满足 L-方程的临界点方程（$n + 2m$ 个方程，$n + 2m$ 个未知数）

条件极值的充分条件

$$H_\mathbf{z} L(\mathbf{z}) = H_\mathbf{z} f(\mathbf{z}) + \Lambda H_\mathbf{z} \pmb{\Phi}(\mathbf{z})$$

定理（条件极值的充分条件）

$$L(\mathbf{z}, \Lambda) = f(\mathbf{z}) + \Lambda \pmb{\Phi}(\mathbf{z})$$

$$A := H_\mathbf{z} L(P, \Lambda) = H_\mathbf{z} L(\mathbf{z}) = H_\mathbf{z} f(P) + \Lambda H_\mathbf{z} \pmb{\Phi}(P)$$

1. 若 $A$ 正定，则 $f(P)$ 为严格条件极小值；
2. 若 $A$ 负定，则 $f(P)$ 为严格条件极大值。

：与无条件极值问题不同，在 $A$ 不定时 $f(P)$ 也有可能取得条件极值。