包络定理

无约束的情形

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]

证明

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。

有约束的情形

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${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial x_{i}}}=0}$ 。

• 霍特林引理
• 谢泼德引理
• 罗伊恒等式