Necessary Extremum Conditions and an Inverse Function Theorem without a priori Normality Assumptions

The problem of minimizing a smooth functional on a given convex closed cone under finitely many equality- and inequality-type constraints is considered. For this problem, an extremum principle, i.e. first- and second-order necessary conditions, is obtained that makes sense even at abnormal points. The extremum principle is generalized to the case of minimizing sequences. Sufficient conditions for an extremum are obtained, and their relation to the necessary conditions is examined. The extremum principle is applied to derive an inverse function theorem, which remains valid at abnormal points.