Let $q \in \mathbb{N}$ and let $a,b \in \mathbb{N}\cup\{0\}$. Let $U \subset \mathbb{R}^q$ be an [open set](/page/Open%20Set), let $z^* \in U$, and let $F:U\to\mathbb{R}$, $A:U\to\mathbb{R}^a$, and $G:U\to\mathbb{R}^b$ be continuously differentiable maps. When $a=0$, interpret $A$ as the unique map into $\mathbb{R}^0$ and impose no equality constraints; when $b=0$, impose no inequality constraints.
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Suppose that $z^*$ is a local minimizer of $F$ over the feasible set
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\begin{align*}
\mathcal{F}:=\{z \in U : A(z)=0 \text{ and } G_i(z)\leq 0 \text{ for every } i \in \{1,\dots,b\}\}.
\end{align*}
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Thus $A(z^*)=0$ and $G_i(z^*)\leq 0$ for every $i \in \{1,\dots,b\}$.
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For each $j\in\{1,\dots,a\}$ let $A_j:U\to\mathbb{R}$ denote the $j$-th component of $A$, and for each $i\in\{1,\dots,b\}$ let $G_i:U\to\mathbb{R}$ denote the $i$-th component of $G$. Define the active inequality set at $z^*$ by
are linearly independent in $\mathbb{R}^q$, with the convention that an empty family is linearly independent.
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Let $DA_{z^*}:\mathbb{R}^q\to\mathbb{R}^a$ and $DG_{z^*}:\mathbb{R}^q\to\mathbb{R}^b$ denote the total derivatives at $z^*$. Let $DA_{z^*}^{\top}:\mathbb{R}^a\to\mathbb{R}^q$ and $DG_{z^*}^{\top}:\mathbb{R}^b\to\mathbb{R}^q$ denote their transpose linear maps with respect to the Euclidean inner products.
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Then there exist multipliers $\lambda \in \mathbb{R}^a$ and $\mu \in \mathbb{R}^b$ such that