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Lagrange multipliers theorem

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

Lagrange multipliers theorem: Let U⊆RnU\subseteq\mathbb R^n be an , let f:U→Rf:U\to\mathbb R and g:U→Rmg:U\to\mathbb R^m be continuously differentiable with m<nm<n, and let S={x∈U:g(x)=0}S=\{x\in U: g(x)=0\}. Assume x∗∈Sx^\ast\in S is a of ff on SS and that Dg(x∗)Dg(x^\ast) has rank mm. Define the Lagrangian

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

Then there exists λ∗∈Rm\lambda^\ast\in\mathbb R^m such that

DxL(x∗,λ∗)=0andg(x∗)=0.D_xL(x^\ast,\lambda^\ast)=0 \quad\text{and}\quad g(x^\ast)=0.
Equivalent characterizations

Equivalently, x∗x^\ast must satisfy the . Solving these equations produces candidate points for constrained extrema on the given .