Strict Constraint Qualifications and Sequential Optimality Conditions for Constrained Optimization

Sequential optimality conditions for constrained optimization are necessarily satisfied by local minimizers, independently of the fulfillment of constraint qualifications. These conditions support the employment of different stopping criteria for practical optimization algorithms. On the other hand, when an appropriate strict constraint qualification associated with some sequential optimality condition holds at a point that satisfies the sequential optimality condition, such point satisfies the Karush-Kuhn-Tucker conditions. This property defines the concept of strict constraint qualification. As a consequence, for each sequential optimality condition, it is natural to ask for its weakest associated constraint qualification. This problem has been solved in a recent paper for the Approximate Karush-Kuhn-Tucker sequential optimality condition. In the present paper we characterize the weakest strict constraint qualifications associated with other sequential optimality conditions that are useful for defining stopping criteria of algorithms. In addition, we prove all the implications between the new strict constraint qualifications and other (classical or strict) constraint qualifications.