Lagrange Auxilliary Function

Given Objective Function $f$ and a Constraint $g=c$, the lagrange auxilliary function $\mathcal{L}=\mathcal{L}(x_1,x_2,\dots ,x_n,\lambda )=f(x_1,\dots ,x_n)-\lambda [g(x_1,\dots ,x_n)-c]$

Special Condition

When $\nabla \mathcal{L}=0$,