ON THE STABILITY OF SETS DEFINED BY A FINITE NUMBER OF EQUALITIES AND INEQUALITIES

Let a set be defined by a finite number of equalities and inequalities. For smooth data, the condition of Mangasarian and Fromovitz is known to be equivalent to the local stability-in a strong sense-of the set. We study here weaker forms of stability. Namely, we state a condition generalizing the one of Mangasarian and Fromovitz that, for some weak form of stability, is necessary. If the gradients of the equality constraints are linearly independent or if there is no equality constraint, this condition is also sufficient.