Critical Cone

Contains directions from $\mathcal{F}(x^{\ast })$ for which it is not clear from first derivative information alone whether $f$ will increase or decrease.

$$\mathcal{C}(x^{\ast },{\lambda }^{\ast })=\{w\in \mathcal{F}(x^{\ast })|\nabla c_i(x^{\ast }{)}^{T}w=0,\text{ all }i\in \mathcal{A}(x^{\ast })\cap \mathcal{I}\text{with}{\lambda }_i^{\ast }>0\}$$

Or an easier way of writing it

$$w\in \mathcal{C}(x^{\ast },{\lambda }^{\ast })⟺\{\begin{array}{ll}\nabla c_i(x^{\ast }{)}^{T}w=0,& \forall i\in \mathcal{E}\\ \nabla c_i(x^{\ast }{)}^{T}w=0,& \forall i\in \mathcal{A}(x^{\ast })\cap \mathcal{I}\text{ with }{\lambda }_i^{\ast }>0\\ \nabla c_i(x^{\ast }{)}^{T}w\ge 0,& \forall i\in \mathcal{A}(x^{\ast })\cap \mathcal{I}\text{ with }{\lambda }_i^{\ast }=0\end{array}$$

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