For convenience, we define the following function:
$$\b{L}(\b{x}, \b{\lambda}, \b{\mu}) = \b{F}(\b{x}) + \sum_{i} \b{\lambda} \b{H}(\b{x}) + \sum_{ i } \b{\mu}\b{G}(\b{x})$$
The tangent space to the surface defined by active constraints can be written as follows:
$$T(\b{x}^*) = \set{ \b{y} \in \R^n : D \b{h}(\b{x}^*)\b{y} = \b{0}, D g_j(\b{x}^*)\b{y} = 0, j \in J(\b{x}^*)} $$
Let $\b{x}^* $ be a local minimizer of $f : \R^n \to \R $ subject to $\b{h}(\b{x}) = \b{0}$ , $\b{g}(\b{x}) \leq \b{0}$, $\b{h}: \R^n \to \R^m$, $m \leq n $, $\b{g} : \R^n \to \R^p $, and $f, \b{h}, \b{g} \in \mathcal{ C }^2 $. Suppose that $\b{x}^* $ is regular. Then, there exist $\b{\lambda}^* \in \R^m $ and $\b{\mu}^* \in \R^p $ such that:
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$\b{\mu}^* \geq \b{0} $, $Df(\b{x}^*) + \transpose{ \b{\lambda}^* } D \b{h}(\b{x}^*) + \transpose{ \b{\mu}^* } D \b{g}(\b{x}^*) = \transpose{ \b{0}} $, $\transpose{ \b{\mu}^* } \b{g}(\b{x}^*) = 0 $
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$\forall \b{y} \in T(\b{x}^*), \transpose{ \b{y}} \b{L}(\b{x}^*, \b{\lambda}^*, \b{\mu}^*) \b{y} \geq 0$
We define the following function: $$\tilde T(\b{x}^*, \b{\mu}^*) = \set{\b{y} \in \R^n : D \b{h}(\b{x}^*)\b{y} = \b{0}, D g_i(\b{x}^*)\b{y} = 0, i \in \tilde J(\b{x}^*, \b{\mu}^*)}$$
where $\tilde J(\b{x}^*, \b{\mu}^*) = \set{ i : g_i(\b{x}^*) = 0, \mu_i^* > 0} $.
Notice that $\tilde J(\b{x}^*, \b{\mu}^*) \subset J(\b{x}^*) $, $T(\b{x}^*) \subset \tilde T(\b{x}^*, \b{\mu}^*) $.
Suppose that $ f, \b{g}, \b{h} \in \mathcal{ C }^2 $ and there exist a feasible point $\b{x}^* \in \R^n $ and vectors $\b{\lambda}^* \in \R^m $ and $\b{\mu}^* \in \R^p $ such that:
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$\b{\mu}^* \geq \b{0} $, $Df(\b{x}^*) + \transpose{ \b{\lambda}^* } D \b{h}(\b{x}^*) + \transpose{ \b{\mu}^* } D \b{g}(\b{x}^*) = \transpose{ \b{0}} $, $\transpose{ \b{\mu}^* } \b{g}(\b{x}^*) = 0 $
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$\forall \b{y} \in \tilde T(\b{x}^*, \b{\mu}^*), \b{y} \neq \b{0}, \transpose{ \b{y}} \b{L}(\b{x}^*, \b{\lambda}^*, \b{\mu}^*) \b{y} > 0$