  In [R. Andreani, G. Haeser, L. M. Mito, H. Ram\'irez C., Weak notions of
nondegeneracy in nonlinear semidefinite programming, arXiv:2012.14810, 2020]
the classical notion of nondegeneracy (or transversality) and Robinson's
constraint qualification have been revisited in the context of nonlinear
semidefinite programming exploiting the structure of the problem, namely, its
eigendecomposition. This allows formulating the conditions equivalently in
terms of (positive) linear independence of significantly smaller sets of
vectors. In this paper we extend these ideas to the context of nonlinear
second-order cone programming. For instance, for an $m$-dimensional
second-order cone, instead of stating nondegeneracy at the vertex as the linear
independence of $m$ derivative vectors, we do it in terms of several statements
of linear independence of $2$ derivative vectors. This allows embedding the
structure of the second-order cone into the formulation of nondegeneracy and,
by extension, Robinson's constraint qualification as well. This point of view
is shown to be crucial in defining significantly weaker constraint
qualifications such as the constant rank constraint qualification and the
constant positive linear dependence condition. Also, these conditions are shown
to be sufficient for guaranteeing global convergence of several algorithms,
while still implying metric subregularity and without requiring boundedness of
the set of Lagrange multipliers.
