Assume that $f : \R^n \to \R $ and $\b{h} : \R^n \to \R^m $, $f, \b{h} \in \mathcal{ C }^2 $. The Lagrangian function is defined as follow:
$$l(\b{x}, \b{\lambda}) = f(\b{x}) + \transpose{ \b{\lambda}} \b{h}(\b{x})$$
Let $\b{L}(\b{x}, \b{\lambda}) $ be the Hessian matrix of $l(\b{x}, \b{\lambda}) $ with respect to $\b{x} $:
$$\b{L}(\b{x}, \b{\lambda}) = \b{F}(\b{x}) + \sum_{ i }\b{\lambda}_i\b{H}_i(\b{x})$$
where $\b{F}(\b{x}) $ is the Hessian matrix of $f $ at $\b{x} $ and $\b{H}_k(\b{x}) $ is the Hessian matrix of $h_k $ at $\b{x} $, $k = 1, 2, …, m $.
Let $\b{x}^*$ be a local minimizer of $f : \R^n \to \R $ subject to $\b{h}(\b{x}) = \b{0}$, $\b{h}: \R^n \to \R^m $, $m \leq n $, and $f, \b{h} \in \mathcal{ C }^2 $. Suppose that $\b{x}^* $ is regular. Then, there exists $\lambda^* \in \R^m$ such that:
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$Df(\b{x}^*) + \transpose{ \b{\lambda}^* } D \b{h}(\b{x}^*) = \transpose{ \b{0}}$.
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$\forall \b{y} \in T(\b{x}^*) , \transpose{ \b{y}} \b{L}(\b{x}^*, \b{ \lambda }^*) \b{y} \geq 0$.
Let $\b{x}^*$ be a local minimizer of $f : \R^n \to \R $ subject to $\b{h}(\b{x}) = \b{0}$, $\b{h}: \R^n \to \R^m $, $m \leq n $, and $f, \b{h} \in \mathcal{ C }^2 $. Suppose that $\b{x}^* $ is regular. Then, there exists $\lambda^* \in \R^m$ such that:
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$Df(\b{x}^*) + \transpose{ \b{\lambda}^* } D \b{h}(\b{x}^*) = \transpose{ \b{0}}$.
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$\forall \b{y} \in T(\b{x}^*) , \transpose{ \b{y}} \b{L}(\b{x}^*, \b{ \lambda }^*) \b{y} > 0$.
Then, $\b{x}^* $ is a strict local minimizer of $f $ subject to $\b{h}(\b{x}) = \b{0} $.