拉格朗日乘子法的簡(jiǎn)單數(shù)學(xué)推導(dǎo)

拉格朗日乘子法公式

結(jié)論
  • 原問(wèn)題

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

  • 轉(zhuǎn)換問(wèn)題
    min\mathbf{F(x)}\tag{2}
    其中
    \mathbf{F(x)} = f( \mathbf{x})+\lambda\mathbf{G(x)}\tag{3}

推導(dǎo)過(guò)程

一、 隱函數(shù)
  1. 將自變量 \mathbf{x} 展開(kāi)成向量形式
    \mathbf{x}=(x_0, x_1, x_2, ..., x_n)
    則等式 \mathbf{G(x)} = 0存在隱函數(shù)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. 隱函數(shù)偏導(dǎo)數(shù)
    對(duì)于等式（方程）\mathbf{G(x)} = 0有式(4)的隱函數(shù)，對(duì)其兩邊同時(shí)進(jìn)行求導(dǎo)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、原問(wèn)題的轉(zhuǎn)換

原問(wèn)題(1)結(jié)合等式(4)可以等價(jià)為
\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}
對(duì)式(7)求解，即為
\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}
觀察(6)式，等式中含有共同項(xiàng)\frac{\partial {\mathbf{G}} }{\partial x_0}，式子兩側(cè)同除以共同項(xiàng)，可以變換為
\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}
將(9)式依次帶入(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}
同時(shí)，式(11)可變換為
\frac{\partial {\mathbf{F}} }{\partial x_0}+ \lambda\frac{\partial {\mathbf{G}} }{\partial x_0}= 0 \tag{13}
結(jié)合式(12)(13)，即可等價(jià)于
\nabla {\mathbf{F}} +\lambda \nabla {\mathbf{G}}=0 \tag{14}
意其即為式(3)的最優(yōu)解

證畢。
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