Second-order conditions in extremal problems. The abnormal points

In this paper we study a minimization problem with constraints and obtain first- and second-order necessary conditions for a minimum. Those conditions -as opposed to the known ones -are also informative in the abnormal case. We have introduced the class of 2-normal constraints and shown that for them the "gap" between the sufficient and the necessary conditions is as minimal as possible. It is proved that a 2-normal mapping is generic. ©1998 American Mathematical Society.