Enhanced Optimality Conditions and Exact Penalty Functions

We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the Fritz-John theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, a new property, pseudonormality, provides the connecting link between the classical constraint qualifications and the use of exact penalty functions.