 Contingent tangent cone

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.