Lagrange With Multiple Constraints

Definition

Given multiple constraints:

• $g_1=c_1$
• $g_2=c_2$
• …
• $g_k=c_k$ With $\{\nabla g_1(x_0),\dots ,\nabla g_k(x_0)\}$ being Linearly Independent Let $S$ be the set where all constraints are satisfied with $x_0\in S$ Then, $f|S$ has local extrema $x_0$ $\nabla f(x_0)={\lambda }_1\nabla g_1(x_0)+{\lambda }_2\nabla g_2(x_0)+\cdots +{\lambda }_k\nabla g_k(x_k)$