# 对偶理论

|             **Primal (P)**             |                 **Dual (D)**                |
| :------------------------------------: | :-----------------------------------------: |
|           maximize $$c^T x$$           |              minimize $$y^T b$$             |
| subject to $$Ax \leq b$$, $$x \geq 0$$ | subject to $$y^T A \geq c^T$$, $$y \geq 0$$ |

{% hint style="info" %} <mark style="color:red;">**定理**</mark> 对偶问题的对偶问题即为原问题
{% endhint %}

{% hint style="info" %} <mark style="color:red;">**定理**</mark> 若 $$x$$ 为原问题的任一可行解, $$y$$ 为对偶问题的任一可行解, 则有 $$c^{T}x \leq y^{T}b$$
{% endhint %}

{% hint style="info" %} <mark style="color:red;">**定理**</mark> 如果 $$x$$ 和 $$y$$ 分别为原问题和对偶问题的一个可行解, 且满足 $$c^{T}x = y^{T}b$$, 则它们分别是原问题和对偶问题的最优解
{% endhint %}

### 影子价格

影子价格是指在其它参数不变的前提下, 某个约束的右边项 (如资源配置问题中的可用资源量) 在一个微小的范围内变动一单位时, 导致的最优目标函数值的变动量

在资源配置问题中, 影子价格反映了各种资源在系统内的稀缺程度. 如果资源供给有剩余 (对于非紧约束), 则进一步增加改资源的供应量不会改变最优决策和最优目标函数值, 因此该资源的影子价格为零. 对于紧约束资源, 增加该资源的供应量有可能会改变最优决策, 也可能不会改变最优决策, 因此该资源的影子价格可能为正, 也可能为零

{% hint style="info" %} <mark style="color:red;">**定理**</mark> 影子价格刚好对应于最优对偶解
{% endhint %}


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