Lagrange multiplier condition

Necessary first-order condition for constrained extrema in terms of gradients.

Lagrange multiplier condition

Lagrange multiplier condition: Let $U\subseteq\mathbb R^n$ be an open set . Let $f:U\to\mathbb R$ and $g:U\to\mathbb R^m$ be continuously differentiable with $m<n$ , and consider the constraint set $S=\{x\in U: g(x)=0\}$ . If $x^\ast\in S$ is a local extremum of $f$ restricted to $S$ and $Dg(x^\ast)$ has rank $m$ (so $x^\ast$ is a regular point of $g$ ), then there exists $\lambda\in\mathbb R^m$ such that

Df(x^\ast)=\lambda^{\mathsf T}Dg(x^\ast).

In the common case $m=1$ , this says the gradient vectors satisfy $\nabla f(x^\ast)=\lambda\,\nabla g(x^\ast)$ , so the constrained extremum occurs at a point where $f$ has a critical-point-type condition compatible with the constraint.