# 最优化问题中的拉格朗日对偶

## 强对偶与KKT条件

$p^{\ast} \geq d^{\ast}$证毕。

1. $f_i(\boldsymbol{x}) \leq 0, \; \forall i \in {1, \cdots, m}$
2. $h_j(\boldsymbol{x}) = 0, \; \forall j \in {1, \cdots, n}$
3. $\lambda_i \geq 0, \; \forall i \in {1, \cdots, m}$
4. $\lambda_i f_i(\boldsymbol{x}) = 0, \; \forall i \in {1, \cdots, m}$
5. $\nabla_\boldsymbol{x}f_0(\boldsymbol{x}) + \sum^m_{i=1}\lambda_i \nabla_{\boldsymbol{x}}f_i(\boldsymbol{x}) + \sum^n_{j=1}\nu_j \nabla_{\boldsymbol{x}} h_j(\boldsymbol{x}) = 0$