Optimality conditions in abnormal extremal problems

In this paper we study the minimization problem with constraints. The first- and second-order necessary conditions are obtained. These conditions are also informative in the abnormal case in contrast to the classical ones. We introduce the class of 2-normal constraints. It is shown that for 2-normal constraints the gap between the obtained necessary conditions and the sufficient conditions is minimal. It is proved that a generic map is 2-normal.