Equality constraints

Here, we consider an optimization problem of the form where and are continuously differentiable.

Let .

Let be the matrix with rows , that is,

A feasible point is regular whenever is of full row rank.

Suppose is a regular point and a local optimal solution. Then there does not exist a vector satisfying

Suppose is a regular point and a local optimal solution. Then there exists a vector such that Proof:
From the previous Lemma, there is no vector satisfying and . This means no vector can satisfy and either, since then would break the Lemma. Thus whenever we have . Thus meaning there exists a with