A second-order sequential optimality condition associated to the convergence of optimization algorithms

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.