On strong and weak second-order necessary optimality conditions for nonlinear programming

Second-order necessary optimality conditions play an important role in optimization theory. This is explained by the fact that most numerical optimization algorithms reduce to finding stationary points satisfying first-order necessary optimality conditions. As a rule, optimization problems, especially the high dimensional ones, have a lot of stationary points so one has to use second-order necessary optimality conditions to exclude nonoptimal points. These conditions are closely related to second-order constraint qualifications, which guarantee the validity of second-order necessary optimality conditions. In this paper, strong and weak second-order necessary optimality conditions are considered and their validity proved under so-called critical regularity condition at local minimizers.