Lagrange multipliers theorem

Constrained extrema give critical points of a Lagrangian under a regularity hypothesis.

Lagrange multipliers theorem

Lagrange multipliers theorem: 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 let $S=\{x\in U: g(x)=0\}$ . Assume $x^\ast\in S$ is a local extremum of $f$ on $S$ and that $Dg(x^\ast)$ has rank $m$ . Define the Lagrangian

L(x,\lambda)=f(x)-\sum_{i=1}^m \lambda_i g_i(x).

Then there exists $\lambda^\ast\in\mathbb R^m$ such that

D_xL(x^\ast,\lambda^\ast)=0 \quad\text{and}\quad g(x^\ast)=0.

Equivalently, $x^\ast$ must satisfy the Lagrange multiplier condition . Solving these equations produces candidate points for constrained extrema on the given constraint set .