Critical cone

The critical cone consists of linearized feasible directions along which the objective function has zero first-order change. For constraints $c_i\ge 0$, its directions satisfy $D{c}_{i}d\ge 0$ at active constraints and $\nabla f^{T}d=0$; equality constraints impose zero linearized residual. Under the usual regularity conditions, second-order Karush-Kuhn-Tucker conditions test the optimization Lagrangian Hessian matrix on this cone. Positive active multipliers force their corresponding directional residuals to vanish.