Lagrange multipliers theorem

Lagrange multipliers theorem: Let $U\subseteq\mathbb{R}^n$ be open, let $f:U\to\mathbb{R}$ and $g:U\to\mathbb{R}^m$ be $C^1$, and define the constraint set $C=\{x\in U: g(x)=0\}.$ Suppose $x^\ast\in C$ is a local maximizer or minimizer of $f$ restricted to $C$, and assume the constraint is regular at $x^\ast$: $\operatorname{rank} Dg(x^\ast)=m.$ Then there exists $\lambda\in\mathbb{R}^m$ such that $\nabla f(x^\ast)=Dg(x^\ast)^{\mathsf T}\lambda.$

This theorem explains the “gradient is normal to the constraint” principle and is a core technique in constrained optimization and geometry. See also Lagrange multiplier condition .

Proof sketch: By the implicit function theorem and the rank hypothesis, the constraint set $C$ is locally a smooth $(n-m)$-dimensional graph. Restrict $f$ to that local parameterization to obtain an unconstrained function of $n-m$ variables; at an interior extremum its gradient vanishes. Translating that condition back to the original coordinates yields the existence of $\lambda$ with the stated relation.