Constraint Qualifications for Semi-Infinite Systems of Convex Inequalities

We introduce and study the Abadie constraint qualification, the weak Pshenichnyi--Levin--Valadier property, and related constraint qualifications for semi-infinite systems of convex inequalities and linear inequalities. Our main results are new characterizations of various constraint qualifications in terms of upper semicontinuity of certain multifunctions. Also, we give some applications of constraint qualifications to linear representations of convex inequality systems, to convex Farkas--Minkowski systems, and to formulas for the distance to the solution set. Some of our concepts and results are new even in the particular case of finite inequality systems.