Let $E$ be a finite-dimensional normed real [vector space](/page/Vector%20Space), let $k \in \mathbb{N}$, let $U \subset E$ be an [open set](/page/Open%20Set), and let $e_0 \in U$. Let
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\begin{align*}
J_0: U \to \mathbb{R}
\end{align*}
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and
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\begin{align*}
G_0: U \to \mathbb{R}^k
\end{align*}
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be $C^1$ maps such that $G_0(e_0)=0$. Suppose that there exists an open neighbourhood $V \subset U$ of $e_0$ such that
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\begin{align*}
J_0(e_0) \leq J_0(e)
\end{align*}
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for every $e \in V$ satisfying $G_0(e)=0$.
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Define the linear maps
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\begin{align*}
j: E \to \mathbb{R}, \qquad j(h) = D(J_0)_{e_0}(h)
\end{align*}
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and
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\begin{align*}
g: E \to \mathbb{R}^k, \qquad g(h) = D(G_0)_{e_0}(h).
\end{align*}
for every $h \in E$. Equivalently, the linear functional $\lambda_0 D(J_0)_{e_0} + \lambda \circ D(G_0)_{e_0}$ on $E$ is the zero functional.
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If, in addition, $D(G_0)_{e_0}: E \to \mathbb{R}^k$ is surjective, then the multiplier pair may be chosen with $\lambda_0=1$: there exists $\lambda \in (\mathbb{R}^k)^*$ such that
for every $h \in E$. A multiplier pair with $\lambda_0>0$ is called normal after rescaling to $\lambda_0=1$; a nonzero multiplier pair with $\lambda_0=0$ is called abnormal.