Second-order conditions in extremal problems with finite-dimensional range. 2-normal maps

A minimization problem with constraints that includes problems for which the constraints are of equality and inequality type is considered. First- and second-order necessary conditions in the Lagrangian form are obtained for this problem. The main difference between these conditions and most of the previously known ones is the fact that they also remain meaningful for abnormal problems, in both the finite-dimensional and infinite-dimensional cases. The notion of 2-normal map is introduced. It is proved that if the map that defines a constraint is 2-normal, then the necessary conditions obtained turn into second-order sufficient conditions after an arbitrarily small perturbation of the problem by terms of second order of smallness. It is also proved that in the space of smooth maps, the set of 2-normal maps is everywhere dense in the Whitney topology.