拉格朗日对偶性(Lagrange duality)
原始问题:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113211524380-543298170.png)
引入拉格朗日函数(Lagrange Function):
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113211844548-1401464763.png)
这里, ![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113212102488-1087717451.png), ![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113212127648-1668157329.png) 是拉格朗日乘子,![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113212232548-651034859.png)。
定义:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113212437849-503774093.png)
极小极大问题为:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113212907550-239901172.png)
定义极小极大问题的最优值:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113212955410-1718233131.png)
**对偶问题**
定义:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113213507778-3015926.png)
极大极小问题为:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113213812579-961154261.png)
定义极大极小问题:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113213957274-1552936354.png)
**原始问题和最优问题的关系**
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113214200089-1853973839.png)
证明如下:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113215136021-629968253.png)
得到最优解如下:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113215312443-975284118.png)
最优解要满足如下的KKT条件:
![](https://img2018.cnblogs.com/blog/1612966/201911/1612966-20191113220047312-836384047.png)