包络定理是带参数的最优化问题中的一个定理。这个定理的内容是，参数的值变动时，目标函数的变动只和参数的变动有关，而与自变量（因参数变动而引起）的变动无关。包络定理在最优化领域非常有用。

设${\displaystyle f(\mathbf {x} ,{\boldsymbol {\alpha )))}$是${\displaystyle \mathbb {R} ^{n+l))$上的可微实函数，其中${\displaystyle \mathbf {x} \in \mathbb {R} ^{n))$是自变量，${\displaystyle {\boldsymbol {\alpha ))\in \mathbb {R} ^{l))$是参数，目标是选择适当的${\displaystyle \mathbf {x} }$以最大化/最小化${\displaystyle f}$。设${\displaystyle V({\boldsymbol {\alpha )))=f(\mathbf {x} ^{*},{\boldsymbol {\alpha )))}$，其中${\displaystyle \mathbf {x} ^{*))$为${\displaystyle f}$取最大值/最小值时的${\displaystyle \mathbf {x} }$，则包络定理即

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

根据全微分公式有

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

因为${\displaystyle \mathbf {x} }$取最值时必有${\displaystyle f}$对${\displaystyle x_{i))$的一阶偏导数为零，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。

在无约束的情形下加上${\displaystyle m}$个同样可微的实约束函数${\displaystyle g_{j}(\mathbf {x} ,{\boldsymbol {\alpha )))=0}$，则包络定理变为

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

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

证明过程与无约束时类似，只是${\displaystyle \mathbf {x} }$取最值时${\displaystyle {\frac {\partial f}{\partial x_{i))}=0}$变为${\displaystyle {\frac {\partial {\mathcal {L))}{\partial x_{i))}=0}$。

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