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  • 拉格朗日对偶性(Lagrange duality)

    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)
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  • 原文地址:https://www.cnblogs.com/hichens/p/11853574.html
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