Definition

Point $x^{n}=(x_{1},\ldots,x_{n})$ and multipliers $\lambda^{m}=(\lambda_{1},\ldots,\lambda_{m})$ and $v^{l}=(v_{1},\ldots,v_{l})$ are said to satisfy the KKT conditions if $$\begin{cases} g_{i}(x^{n})\le0, \ \lambda_{i}\ge0, \ \lambda_{i}g_{i}(x^{n})=0&\mbox{for }i=1,\ldots,m \\ h_{j}(x^{n})=0&\mbox{for }j=1,\ldots,l \\ \frac{\partial f(x^{n})}{\partial x_{K}}+\sum\limits^{m}_{i=1}\lambda_{i}\frac{\partial g_{i}(x^{n})}{\partial x_{K}}+\sum\limits^{l}_{j=1}v_{j}\frac{\partial h_{j}(x^{n})}{\partial x_{K}}=0&\mbox{for }K=1,\ldots,n \end{cases}$$