Contingent tangent cone ($T_{\Omega }(x)=\{{\lim }_n(x_n-x)/t_n:x_n\in \Omega ,t_n\downarrow 0\}$)

This cone records first-order limiting velocities of feasible points approaching $x$. The defining limits are required to exist, and the point sequences must be feasible. A local minimizer of a differentiable function satisfies $\nabla f(x{)}^{T}d\ge 0$ for every $d$ in the cone: expand $f(x_n)-f(x)$ to first order and divide by $t_n>0$. A tangent direction need not itself generate an exactly feasible straight segment. Linearizing active constraints always gives a containing linearized tangent cone; equality requires a suitable constraint qualification.

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.