= Linearized tangent cone
{title2=$F(x)=\{d:\nabla c_j(x)^Td\geq0\ (j\in\mathcal A(x))\}$}
For smooth inequalities $c_j\geq0$, the displayed intersection of half-spaces contains the <contingent tangent cone>. This follows by differentiating each active feasibility inequality along a feasible tangent sequence. The <linear independence constraint qualification>, and more generally the <Mangasarian–Fromovitz constraint qualification>, ensure equality in the inequality-only setting. Without a qualification the inclusion can be strict: for $c(x)=-x^2\geq0$, the feasible set is $\{0\}$ and its tangent cone is $\{0\}$, while $c'(0)=0$ makes the linearized cone the whole real line.
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