Nondegenerate Constraint Qualification¶

Definition¶

Write the constraint map as $\boldsymbol{h}=(h_1,\ldots,h_m)$ $(h_{1}, \dots, h_{m})$ . For equality constraints $h_1(\boldsymbol{x})=c_1,\ldots,h_m(\boldsymbol{x})=c_m$ $c_{m}$ , the nondegenerate constraint qualification is satisfied at $\boldsymbol{x}^\ast$ if

\operatorname{rank}(\boldsymbol{D}\boldsymbol{h}(\boldsymbol{x}^\ast))=m.

With inequality constraints, the same rank condition is applied to the gradients of the binding inequality constraints together with the equality constraints.

Interpretation¶

NDCQ says the active constraints are locally independent. If it fails, some constraint may be redundant or the feasible set may have a singular shape.

Two panels of two constraint curves. Left: the curves cross and their gradients are independent, so the qualification holds. Right: the curves touch and their gradients are parallel, so it fails.

On the left, the constraints cross transversally and $\operatorname{rank}\boldsymbol{D}\boldsymbol{h}(\boldsymbol{x}^\ast)=2=m$ $m$ , so the qualification holds. On the right, they touch with parallel gradients and $\operatorname{rank}\boldsymbol{D}\boldsymbol{h}(\boldsymbol{x}^\ast)=1<m$ $m$ , so it fails.

The Jacobian collects the constraint gradients as rows. Its rank equals the number of active constraints exactly when these rows are independent, excluding a nonzero linear relation between their first-order effects.

Key Results¶

When NDCQ holds, the regular Lagrange necessary conditions apply at a constrained local optimum.
If NDCQ fails, the point remains a candidate that must be checked separately.
For inequality problems, nonbinding inequalities are not included in the active rank check.

Examples¶

For $h(x,y)=x+y$ $y$ , NDCQ holds at every point because the gradient is nonzero.
For $h_1(x,y)=x+y$ and $h_2(x,y)=2x+2y$ $2 y$ , NDCQ fails because the gradients are dependent.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 4, Lecture 2
Math IV: Multivariate Calculus · Week 5, Lecture 1
Math IV: Multivariate Calculus · Week 5, Lecture 2

Aliases: NDCQ, LICQ Keywords: constraint qualification, active constraint rank condition, linear independence constraint qualification

Last updated: 26 October 2026 00:09