On the Necessity of some Constraint Qualification Conditions in Convex Programming

We realize a study of various constraint qualification conditions for the existence of Lagrange multipliers for convex minimization problems in general normed vector spaces; it is based on a new formula for the normal cone to the constraint set, on local metric regularity and a metric regularity property on bounded subsets. As a by-product we obtain a characterization of the metric regularity of a finite family of closed convex sets.