Linearized tangent cone ($F(x)=\{d:\nabla c_j(x{)}^{T}d\ge 0(j\in \mathcal{A}(x))\}$)

For smooth inequalities $c_j\ge 0$, 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 $c(x)=-x^2\ge 0$, the feasible set is $\{0\}$ and its tangent cone is $\{0\}$, while $c^{\prime }(0)=0$ makes the linearized cone the whole real line.