包絡定理

包絡定理(Envelop Theorem)是帶參數的最佳化問題中的一個定理。這個定理的內容是，參數的值變動時，目標函數的變動只和參數的變動有關，而與自變數（因參數變動而引起）的變動無關。包絡定理在最佳化領域非常有用。

具體表述

無約束的情形

設${\displaystyle f(\mathbf {x} ,{\boldsymbol {\alpha }})}$是${\displaystyle \mathbb {R} ^{n+l}}$上的可微實函數，其中${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$是自變數，${\displaystyle {\boldsymbol {\alpha }}\in \mathbb {R} ^{l}}$是參數，目標是選擇適當的${\displaystyle \mathbf {x} }$以極大化/極小化${\displaystyle f}$。設${\displaystyle V({\boldsymbol {\alpha }})=f(\mathbf {x} ^{*},{\boldsymbol {\alpha }})}$，其中${\displaystyle \mathbf {x} ^{*}}$為${\displaystyle f}$取最大值/最小值時的${\displaystyle \mathbf {x} }$，則包絡定理即

${\displaystyle {\frac {\mathrm {d} V}{\mathrm {d} {\boldsymbol {\alpha }}}}=\left.{\frac {\partial f}{\partial {\boldsymbol {\alpha }}}}\right\vert _{\mathbf {x} =\mathbf {x} ^{*}}}$。^[1][2]

證明

根據全微分公式有

${\displaystyle \mathrm {d} V=\left.\mathrm {d} f\right\vert _{\mathbf {x} =\mathbf {x} ^{*}}=\left.\left(\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\mathrm {d} x_{i}+\sum _{i=1}^{l}{\frac {\partial f}{\partial \alpha _{i}}}\mathrm {d} \alpha _{i}\right)\right\vert _{\mathbf {x} =\mathbf {x} ^{*}}}$。

因為${\displaystyle \mathbf {x} }$取最值時必有${\displaystyle f}$對${\displaystyle x_{i}}$的一階偏導數為零，即

${\displaystyle \left.{\frac {\partial f}{\partial \mathbf {x} }}\right\vert _{\mathbf {x} =\mathbf {x} ^{*}}=0}$，

故可得到

${\displaystyle \mathrm {d} V=\left.\sum _{i=1}^{n}{\frac {\partial f}{\partial \alpha _{i}}}\mathrm {d} \alpha _{i}\right\vert _{\mathbf {x} =\mathbf {x} ^{*}}}$，

也即${\displaystyle {\frac {\mathrm {d} V}{\mathrm {d} {\boldsymbol {\alpha }}}}=\left.{\frac {\partial f}{\partial {\boldsymbol {\alpha }}}}\right\vert _{\mathbf {x} =\mathbf {x} ^{*}}}$成立。

有約束的情形

在無約束的情形下加上${\displaystyle m}$個同樣可微的實約束函數${\displaystyle g_{j}(\mathbf {x} ,{\boldsymbol {\alpha }})=0}$，則包絡定理變為

${\displaystyle {\frac {\mathrm {d} V}{\mathrm {d} {\boldsymbol {\alpha }}}}=\left.{\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\alpha }}}}\right\vert _{\mathbf {x} =\mathbf {x} ^{*}}}$，

其中${\displaystyle {\mathcal {L}}(\mathbf {x} ,{\boldsymbol {\lambda }},{\boldsymbol {\alpha }})=f(\mathbf {x} ,{\boldsymbol {\alpha }})+\sum _{j=1}^{m}\lambda _{j}({\boldsymbol {\alpha }})g_{j}(\mathbf {x} ,{\boldsymbol {\alpha }})}$是拉格朗日函數。

證明過程與無約束時類似，只是${\displaystyle \mathbf {x} }$取最值時${\displaystyle {\frac {\partial f}{\partial x_{i}}}=0}$變為${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial x_{i}}}=0}$。