= Contingent tangent cone
{title2=$T_\Omega(x)=\{\lim_n(x_n-x)/t_n:x_n\in\Omega,\ t_n\downarrow0\}$}
= Tangent cone in optimization
{synonym}
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 $\nabla f(x)^Td\geq0$ 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>.
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