Constraint qualifications for semi-infinite systems of convex inequalities

We introduce and study the Abadie constraint qualification, the weak Psheni chnyi-Levin-Valadier property, and related constraint qualifications for se mi-infinite systems of convex inequalities and linear inequalities. Our mai n results are new characterizations of various constraint qualifications in terms of upper semicontinuity of certain multifunctions. Also, we give som e applications of constraint qualifications to linear representations of co nvex inequality systems, to convex Farkas-Minkowski systems, and to formula s for the distance to the solution set. Some of our concepts and results ar e new even in the particular case of finite inequality systems.