A second-order optimality condition with first and second-order complementarity associated to global convergence of algorithms

We develop a new notion of second-order complementarity with respect to the tangent subspace associated to second-order necessary optimality conditions by the introduction of so-called tangent multipliers. We prove that around a local minimizer, a second-order stationarity residual can be driven to zero while controlling the growth of Lagrange multipliers and tangent multipliers, which gives a new strong second-order optimality condition without constraint qualifications. We prove that second-order variants of augmented Lagrangian and interior point methods satisfy our optimality condition. Finally, we present a companion minimal constraint qualification, weaker than the ones known for second-order methods, that ensures usual global convergence results to a classical second-order stationary point.