  Second-order necessary optimality conditions for nonlinear conic programming
problems that depend on a single Lagrange multiplier are usually built under
nondegeneracy and strict complementarity. In this paper we establish a
condition of such type for two classes of nonlinear conic problems, namely
semidefinite and second-order cone programming, assuming Robinson's constraint
qualification and a weak constant rank-type property which are, together,
strictly weaker than nondegeneracy. Our approach is done via a penalty-based
strategy, which is aimed at providing strong global convergence results for
first- and second-order algorithms. Since we are not assuming strict
complementarity, the critical cone does not reduce to a subspace, thus, the
second-order condition we arrive at is defined in terms of the lineality space
of the critical cone. In the case of nonlinear programming, this condition
reduces to the standard second-order condition widely used as second-order
stationarity measure in the algorithmic practice.
