机器学习7 拉格朗日乘数法

来源：互联网发布：智能网络液晶平板电视编辑：程序博客网时间：2024/04/29 14:28

{z = f (x, y) h (x, y) = 0} \to min f (x, y) h (x, y) = 0 \Rightarrow y = φ (x) \partial h ( x , y ) \partial x + \partial h ( x , y ) \partial y d ( φ ( x ) ) d ( x ) = 0 f (x, y) = f (x, φ (x)) \Rightarrow \partial f ( x , y ) \partial x + \partial f ( x , y ) \partial y d ( φ ( x ) ) d ( x ) = 0 \Leftrightarrow ⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h (x, y) = 0 \partial h ( x , y ) \partial x + \partial h ( x , y ) \partial y d ( φ ( x ) ) d ( x ) = 0 \partial f ( x , y ) \partial x + \partial f ( x , y ) \partial y d ( φ ( x ) ) d ( x ) = 0 \Leftrightarrow ⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h (x, y) = 0 \partial f ( x , y ) \partial x \partial h ( x , y ) \partial x = \partial f ( x , y ) \partial y \partial h ( x , y ) \partial y = - λ \Leftrightarrow ⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h (x, y) = 0 \partial f ( x , y ) \partial x + λ \partial h ( x , y ) \partial x = 0 \partial f ( x , y ) \partial y + λ \partial h ( x , y ) \partial y = 0 L (x, y, λ) = f (x, y) + λ h (x, y) \Leftrightarrow ⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ \partial L ( x , y ) \partial λ = 0 \partial L ( x , y ) \partial x = 0 \partial L ( x , y ) \partial y = 0

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