 Solution

For $\mathcal{L}=f-y^{T}c$, the inequality-constrained Karush-Kuhn-Tucker conditions areUnder an appropriate constraint qualification, the second-order necessary condition is $d^{T}H(x_{\ast },y_{\ast })d\ge 0$ on the critical coneEquivalently active constraints with positive multiplier require $A_{i}d=0$, while active zero-multiplier constraints retain $A_{i}d\ge 0$. Inactive constraints impose no first-order tangent restriction. This distinction is why the relevant cone need not be the nullspace of every active row.