  The constraint nondegeneracy condition is one of the most relevant and useful
constraint qualifications in nonlinear semidefinite programming. It can be
characterized in terms of any fixed orthonormal basis of the, let us say,
$\ell$-dimensional kernel of the constraint matrix, by the linear independence
of a set of $\ell(\ell+1)/2$ derivative vectors. We show that this linear
independence requirement can be equivalently formulated in a smaller set, of
$\ell$ derivative vectors, by considering all orthonormal bases of the kernel
instead. This allows us to identify that not all bases are relevant for a
constraint qualification to be defined, giving rise to a strictly weaker
variant of nondegeneracy related to the global convergence of an external
penalty method. Also, by exploiting the sparsity structure of the constraints,
we were able to define another weak variant of nondegeneracy by removing the
null entries from consideration. In particular, both our new constraint
qualifications reduce to the linear independence constraint qualification for
nonlinear programming when considering a diagonal semidefinite constraint. More
generally, when the problem has a diagonal block structure, the conditions
formulated as a single block diagonal matrix constraint are equivalent to their
analogues formulated with several semidefinite matrices.
