Abstract. We introduce extensions of the Mangasarian-Fromovitz and the Abadie constraint qualifications to non-smooth optimization problems. We do not assume directional differentiability but only upper semi-continuity of the defining functions. By deriving and reviewing primal first order optimality conditions for non-smooth problems we motivate the formulations of the constraint qualifications, we study their interrelation, and we show how they are related to Slater's condition for non-smooth convex problems, to non-smooth reverse-convex problems, and to the stability of parametric feasible set mappings.
The appropriate use of alternative theorems to derive dual optimality conditions is briefly highlighted. In the literature on general semi-infinite programming problems a number of formally different extensions of the Mangasarian-Fromovitz constraint qualifications have recently been introduced under different structural assumptions. We show that all these extensions are unified by the constraint qualification presented here.