查看“︁包络定理”︁的源代码

{{NoteTA |G1=Economics|G2=Math}}
{{expand|time=2018-11-11T03:11:01+00:00}}
'''包络定理'''(Envelop Theorem)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。

== 具体表述 ==
=== 无约束的情形 ===
设<math>f(\mathbf{x}, \boldsymbol{\alpha})</math>是<math>\mathbb{R}^{n+l}</math>上的[[可微]]实函数，其中<math>\mathbf{x} \in \mathbb{R}^{n}</math>是自变量，<math>\boldsymbol{\alpha} \in \mathbb{R}^{l}</math>是参数，目标是选择适当的<math>\mathbf{x}</math>以最大化/最小化<math>f</math>。设<math>V(\boldsymbol{\alpha}) = f(\mathbf{x}^*, \boldsymbol{\alpha})</math>，其中<math>\mathbf{x}^*</math>为<math>f</math>取最大值/最小值时的<math>\mathbf{x}</math>，则包络定理即
:<math>\frac{\mathrm{d} V}{\mathrm{d} \boldsymbol{\alpha}} = \left.\frac{\partial f}{\partial \boldsymbol{\alpha}}\right\vert_{\mathbf{x} = \mathbf{x}^*}</math>。<ref name="Afriat, 1971">{{cite journal |first=S. N. |last=Afriat |year=1971 |title=Theory of Maxima and the Method of Lagrange |journal=[[SIAM Journal on Applied Mathematics]] |volume=20 |issue=3 |pages=343–357 |doi=10.1137/0120037 }}</ref><ref>{{cite book |first=Akira |last=Takayama |title=Mathematical Economics |location=New York |publisher=Cambridge University Press |edition=Second |year=1985 |isbn=0-521-31498-4 |pages=137–138 |url=https://books.google.com/books?id=j6PLOBFotPQC&pg=PA137 |access-date=2018-11-10 |archive-date=2017-02-22 |archive-url=https://web.archive.org/web/20170222124147/https://books.google.com/books?id=j6PLOBFotPQC&pg=PA137 |dead-url=no }}</ref>

==== 证明 ====
根据[[全微分]]公式有
:<math>\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}^*}</math>。
因为<math>\mathbf{x}</math>取最值时必有<math>f</math>对<math>x_i</math>的一阶偏导数为零，即
:<math>\left.\frac{\partial f}{\partial \mathbf{x}}\right\vert_{\mathbf{x}=\mathbf{x}^*} = 0</math>，
故可得到
:<math>\mathrm{d}V = \left.\sum_{i=1}^n\frac{\partial f}{\partial \alpha_i}\mathrm{d}\alpha_i\right\vert_{\mathbf{x}=\mathbf{x}^*}</math>，
也即<math>\frac{\mathrm{d} V}{\mathrm{d} \boldsymbol{\alpha}} = \left.\frac{\partial f}{\partial \boldsymbol{\alpha}}\right\vert_{\mathbf{x} = \mathbf{x}^*}</math>成立。

=== 有约束的情形 ===
在无约束的情形下加上<math>m</math>个同样可微的实约束函数<math>g_j(\mathbf{x}, \boldsymbol{\alpha})=0</math>，则包络定理变为
:<math>\frac{\mathrm{d} V}{\mathrm{d} \boldsymbol{\alpha}} = \left.\frac{\partial \mathcal{L}}{\partial \boldsymbol{\alpha}}\right\vert_{\mathbf{x} = \mathbf{x}^*}</math>，
其中<math>\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})</math>是[[拉格朗日乘数|拉格朗日函数]]。

证明过程与无约束时类似，只是<math>\mathbf{x}</math>取最值时<math>\frac{\partial f}{\partial x_i} = 0</math>变为<math>\frac{\partial \mathcal{L}}{\partial x_i} = 0</math>。

== 参考文献 ==
{{reflist}}

== 参见 ==
*[[霍特林引理]]
*[[谢泼德引理]]
*[[罗伊恒等式]]

[[Category:生产经济学]]
[[Category:变分法]]
[[Category:经济学定理]]
[[Category:分析定理]]