Active constraint set ($\mathcal{A}(x)=\{j:c_j(x)=0\}$)

For inequalities $c_j(x)\ge 0$, the active constraints at a feasible point are those holding with equality. Inactive constraints remain strictly satisfied in a sufficiently small neighborhood and do not restrict first-order feasible velocities there. The active gradients determine the linearized tangent cone and appear in the linear independence constraint qualification. At a KKT point, inactive constraints have zero Lagrange multipliers by complementary slackness.