目录：
一般约束问题
形式：
$$
\begin{cases}
min\,\,f\left( x \right) ,\,\,x\in R^n\
s.t.\,\,\begin{matrix}
g_i\left( x \right) =0,\,\,j=1,\cdots ,m\
h_k\left( x \right) \le 0,\,\,k=1,\cdots ,p\
\end{matrix}\
\end{cases}
$$
定义Lagrangian函数
$$
L\left( x,\left{ \lambda j \right} ,\left{ u_k \right} \right) =f\left( x \right) +\sum{j=1}^m{\lambda jg_j\left( x \right)}+\sum{k=1}^p{\mu _kh_k\left( x \right)}
$$
Karush-Kuhn-Tucker条件：
$$
\begin{cases}
\nabla xL=\nabla _xf\left( x \right) +\sum{j=1}^m{\lambda j\nabla g_j\left( x \right) +\sum{k=1}^p{\mu _k\nabla _kh\left( x \right)}}=0\,\, \left( Stationarity \right)\
g_j\left( x \right) =0,j=1,\cdots ,m\
h_k\left( x \right) \le 0\,\, \left( Primal\,\,Feasibility\text{，原始可行性} \right)\
\mu _k\ge 0\,\, \left( Dual\,\,Feasibility\text{，对偶可行性} \right)\
\mu _kh_k\left( x \right) =0,k=1,\cdots ,p\,\, \left( Complementary\,\,Slackness \right)\
\end{cases}
Original: https://www.cnblogs.com/zhaoke271828/p/16739942.html
Author: Viktor_Cullen
Title: 最优化-约束优化问题的最优性条件
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