A practical constraint qualification

Constraint qualification 1.1 is not easily verifiable, although it is just what is needed to prove the existence of Lagrange multipliers. I now present another constraint qualification, stronger than 1.1, that is easier to verify.

Constraint qualification 4.1   The matrix $\nabla g(x^*)$ has full rank, that is, the columns of $\nabla g(x^*)$ are linearly independent.

A point x^* satisfying constraint qualification 4.1 is said to be a regular point of the feasible set defined by g(x)=0 or of the nonlinear program (1).

I will now prove that constraint qualification (1.1) is satisfied at every regular point of g(x)=0. The proof is fairly simple in outline, although the notation is rather difficult to follow. I will need some results concerning linear least-squares problems.

Theorem 4.2   Suppose $A\in{\bf {\rm R}}^{m\times n}$, where $m\ge n$, has full rank. Then, for any $b\in{\bf {\rm R}}^m$, the least-squares problem

$$\min\Vert Ax-b\Vert$$ (11)

has a unique solution, given by

$$x=\left(A^TA\right)^{-1}A^Tb.$$