拉格朗日乘子法的简单数学推导

拉格朗日乘子法公式

结论

原问题

转换问题

其中

推导过程

一、隐函数

将自变量展开成向量形式

则等式存在隐函数使得

令
隐函数偏导数
对于等式（方程）有式的隐函数，对其两边同时进行求导得
$\frac{\partial {\mathbf{G_{x'}}}}{\partial x_1} = \frac{\partial {\mathbf{G}} }{\partial x_1}+ \frac{\partial {\mathbf{G}} }{\partial x_0} \frac{\partial {{g}} }{\partial x_1} = 0 \\ \frac{\partial {\mathbf{G_{x'}}}}{\partial x_2} = \frac{\partial {\mathbf{G}} }{\partial x_2}+ \frac{\partial {\mathbf{G}} }{\partial x_0} \frac{\partial {{g}} }{\partial x_2} = 0 \\ \ \\ ... \\ \ \\ \frac{\partial {\mathbf{G_{x'}}}}{\partial x_n} = \frac{\partial {\mathbf{G}} }{\partial x_n}+ \frac{\partial {\mathbf{G}} }{\partial x_0} \frac{\partial {{g}} }{\partial x_n} = 0 \tag{6}$

二、原问题的转换

原问题结合等式可以等价为

对式求解，即为
$\frac{\partial {\mathbf{F_{x'}}}}{\partial x_1} = \frac{\partial {\mathbf{F}} }{\partial x_1}+ \frac{\partial {\mathbf{F}} }{\partial x_0} \frac{\partial {{g}} }{\partial x_1} = 0 \\ \frac{\partial {\mathbf{F_{x'}}}}{\partial x_2} = \frac{\partial {\mathbf{F}} }{\partial x_2}+ \frac{\partial {\mathbf{F}} }{\partial x_0} \frac{\partial {{g}} }{\partial x_2} = 0 \\ \ \\ ... \\ \ \\ \frac{\partial {\mathbf{F_{x'}}}}{\partial x_n} = \frac{\partial {\mathbf{F}} }{\partial x_n}+ \frac{\partial {\mathbf{F}} }{\partial x_0} \frac{\partial {{g}} }{\partial x_n} = 0 \tag{8}$
观察式，等式中含有共同项，式子两侧同除以共同项，可以变换为
$\frac{\partial {{g}} }{\partial x_1} = -\frac{\frac{\partial {\mathbf{G}} }{\partial x_1}}{\frac{\partial {\mathbf{G}} }{\partial x_0}} \\ \frac{\partial {{g}} }{\partial x_2} = -\frac{\frac{\partial {\mathbf{G}} }{\partial x_2}}{\frac{\partial {\mathbf{G}} }{\partial x_0}} \\ \ \\ ... \\ \ \\ \frac{\partial {{g}} }{\partial x_n} = -\frac{\frac{\partial {\mathbf{G}} }{\partial x_n}}{\frac{\partial {\mathbf{G}} }{\partial x_0}} \tag{9}$
将式依次带入式，得
$\frac{\partial {\mathbf{F_{x'}}}}{\partial x_1} = \frac{\partial {\mathbf{F}} }{\partial x_1}+ \frac{\partial {\mathbf{F}} }{\partial x_0} (-\frac{\frac{\partial {\mathbf{G}} }{\partial x_1}}{\frac{\partial {\mathbf{G}} }{\partial x_0}})= \frac{\partial {\mathbf{F}} }{\partial x_1}+ (-\frac{\frac{\partial {\mathbf{F}} }{\partial x_0}}{{\frac{\partial {\mathbf{G}} }{\partial x_0}}})\frac{\partial {\mathbf{G}} }{\partial x_1}= 0 \\ \frac{\partial {\mathbf{F_{x'}}}}{\partial x_2} = \frac{\partial {\mathbf{F}} }{\partial x_2}+ \frac{\partial {\mathbf{F}} }{\partial x_0} (-\frac{\frac{\partial {\mathbf{G}} }{\partial x_2}}{\frac{\partial {\mathbf{G}} }{\partial x_0}})= \frac{\partial {\mathbf{F}} }{\partial x_2}+ (-\frac{\frac{\partial {\mathbf{F}} }{\partial x_0}}{{\frac{\partial {\mathbf{G}} }{\partial x_0}}})\frac{\partial {\mathbf{G}} }{\partial x_2}= 0 \\ \ \\ ... \\ \ \\ \frac{\partial {\mathbf{F_{x'}}}}{\partial x_n} = \frac{\partial {\mathbf{F}} }{\partial x_n}+ \frac{\partial {\mathbf{F}} }{\partial x_0} (-\frac{\frac{\partial {\mathbf{G}} }{\partial x_n}}{\frac{\partial {\mathbf{G}} }{\partial x_0}})= \frac{\partial {\mathbf{F}} }{\partial x_n}+ (-\frac{\frac{\partial {\mathbf{F}} }{\partial x_0}}{{\frac{\partial {\mathbf{G}} }{\partial x_0}}})\frac{\partial {\mathbf{G}} }{\partial x_n}= 0 \tag{10}$
令
代入得
$\frac{\partial {\mathbf{F}} }{\partial x_1}+ \lambda\frac{\partial {\mathbf{G}} }{\partial x_1}= 0 \\ \frac{\partial {\mathbf{F}} }{\partial x_2}+ \lambda\frac{\partial {\mathbf{G}} }{\partial x_2}= 0 \\ \ \\ ... \\ \ \\ \frac{\partial {\mathbf{F}} }{\partial x_n}+ \lambda\frac{\partial {\mathbf{G}} }{\partial x_n}= 0 \tag{12}$
同时，式可变换为

结合式，即可等价于

意其即为式的最优解

拉格朗日乘子法的简单数学推导

拉格朗日乘子法公式

结论

推导过程

一、 隐函数

二、原问题的转换

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一、隐函数