Sequential optimality conditions have recently played an important role on the analysis of the global convergence of optimization algorithms towards first-order stationary points, justifying their stopping criteria. In this article, we introduce a sequential optimality condition that takes into account second-order information and that allows us to improve the global convergence assumptions of several second-order algorithms, which is our main goal. We also present a companion constraint qualification that is less stringent than previous assumptions associated to the convergence of second-order methods, like the joint condition Mangasarian–Fromovitz and weak constant rank. Our condition is also weaker than the constant rank constraint qualification. This means that we can prove second-order global convergence of well–established algorithms even when the set of Lagrange multipliers is unbounded, which was a limitation of previous results based on Mangasarian–Fromovitz constraint qualification. We prove global convergence of well-known variations of the augmented Lagrangian and regularized sequential quadratic programming methods to second-order stationary points under this new weak constraint qualification.

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