The Supporting Role of the Mangasarian-Fromovitz Constraint Qualification in Calculus of Variations

The Mangasarian-Fromovitz constraint qualification has played a fundamental role in mathematical programming problems involving inequality constraints. It is known to be equivalent to a normality condition (in terms of the positive linear independence of active gradients) which, in turn, implies regularity (the tangent and the linearizing cones coincide), a condition which has been crucial in the derivation of first- and second-order necessary optimality conditions. In this paper, we study the corresponding implications for problems in the calculus of variations. In particular, we show how the equivalence between normality and the Mangasarian-Fromovitz constraint qualification is preserved, but also that their main role changes completely since, as a simple example shows, they may not imply the corresponding regularity condition.