On the Classical Necessary Second-Order Optimality Conditions

In this paper, the property of a necessary second-order optimality condition to hold with the same Lagrange multiplier for all critical vectors is investigated. It is limited to nonconvex optimization Problems in ⁿ with equality and inequality constraints; the Mangasarian-Fromovitz constraint qualification is assumed to be hold. A counterexample was given recently by Anitescu. We give some sufficient conditions and we prove that this property holds if n 2 or if the number of active inequality constraints is at most two. For three active inequality constraints and n =3, a counterexample is given.