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

min f( \mathbf{x}) \\ st. \mathbf{G(x)} = 0 \tag{1}

  • 转换问题
    min\mathbf{F(x)}\tag{2}
    其中
    \mathbf{F(x)} = f( \mathbf{x})+\lambda\mathbf{G(x)}\tag{3}

推导过程

一、 隐函数
  1. 将自变量 \mathbf{x} 展开成向量形式
    \mathbf{x}=(x_0, x_1, x_2, ..., x_n)
    则等式 \mathbf{G(x)} = 0存在隐函数使得
    x_0=g(x_1,x_2,x_3, ..., x_n) \tag{4}
    令
    \mathbf{x'}=(x_1,x_2,x_3, ..., x_n) \tag{5}
  2. 隐函数偏导数
    对于等式（方程）\mathbf{G(x)} = 0有式(4)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(1)结合等式(4)可以等价为
\begin{align} min\mathbf{F(x)} &= min f(x_0,x_1, x_2,...,x_n)\\ &= minf(g(x_1,x_2,x_3,...,x_n),x_1,x_2,x_3,...x_n)\\ \tag{7}\end{align}
对式(7)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(6)式，等式中含有共同项\frac{\partial {\mathbf{G}} }{\partial x_0}，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(9)式依次带入(8)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\lambda=-\frac{\frac{\partial {\mathbf{F}} }{\partial x_0}}{{\frac{\partial {\mathbf{G}} }{\partial x_0}}}\tag{11}
代入(10)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(11)可变换为
\frac{\partial {\mathbf{F}} }{\partial x_0}+ \lambda\frac{\partial {\mathbf{G}} }{\partial x_0}= 0 \tag{13}
结合式(12)(13)，即可等价于
\nabla {\mathbf{F}} +\lambda \nabla {\mathbf{G}}=0 \tag{14}
意其即为式(3)的最优解

证毕。
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