Second-order necessary or sufficient optimality conditions for nonlinear programming are usually defined by means of the positive (semi-)definiteness of a quadratic form, or a maximum of quadratic forms, over the critical cone. However, algorithms for finding such second-order stationary points are currently unknown. Typically, an algorithm with a second-order property approximates a primal-dual point such that the Hessian of the Lagrangian evaluated at the limit point is, under a constraint qualification, positive semidefinite over the lineality space of the critical cone. This is in general a proper subset of the critical cone, unless one assumes strict complementarity, which gives a weaker necessary optimality condition.
In this paper, a new strong sequential optimality condition is suggested and analyzed. Based on this, we propose a penalty algorithm which, under certain conditions, is able to achieve second-order stationarity with positive semidefiniteness over the whole critical cone, which corresponds to a strong necessary optimality condition. Although the algorithm we propose is somewhat of a theoretical nature, our analysis provides a general framework for obtaining strong second-order stationarity.