# Lagrangian

For the mathematical program

$LaTeX: \min \{ f(x) : g(x) \ge 0, \, h(x) = 0, \, x \in X \},$
the Lagrangian is the function:

$LaTeX: L(x, u, v) = f(x) - u^T g(x) - v^T h(x).$

for $LaTeX: x \in X$ and $LaTeX: u \ge 0$. Note that the Lagrange Multiplier Rule can be written as the first-order conditions for $LaTeX: (x^*, u^*,v^*)$ to be a saddle point of $LaTeX: L$. In Lagrange's multiplier theorem (where $LaTeX: X=\mathbb{R}^n$ and $LaTeX: g$ is vacuous), this is simply that $LaTeX: \nabla L(x^*,v^*)=0$, which could be any type of stationary point.