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

Global Convergence of Algorithms Under Constant Rank Conditions for Nonlinear Second-Order Cone Programming