Locally Farkas-Minkowski systems in convex semi-infinite programming

A pair of constraint qualifications in convex semi-infinite programming, na mely the locally Farkas-Minkowski constraint qualification and generalized Slater constraint qualification, are studied in the paper. We analyze the r elationship between them, as well as the behavior of the so-called active a nd sup-active mappings, accounting for the tightness of the constraint syst em at each point of the variables space. The generalized Slater constraint qualification guarantees a regular behavior of the supremum function (defin ed as supremum of the infinitely many functions involved in the constraint system), giving rise to the well-known Valadier formula.