Asymptotic Constraint Qualifications and Error Bounds for Semi-Infinite Systems of Convex Inequalities

We extend the known asymptotic constraint qualifications (ACQs) and some related constants from finite to semi-infinite convex inequality systems. We show that, in contrast with the finite case, only some of these ACQs are equivalent and only some of these constants coincide, unless we assume the”weak Pshenichnyi-Levin-Valadier property” introduced in [12]. We extend most of the global error bound results of [10] from finite systems of convex inequalities to the semi-infinite case and we show that to each semi-infinite convex inequality system with fini te-valued”sup function” one can associate an equivalent semi-infinite convex inequality system with finite-valued sup-function, admitting a global error bound. We give examples that the classical theorem of Hoffman [8] on the existence of a global error bound for each finite linear inequality system, as well as a result of [11] on global error bounds for finite differend able convex inequality systems cannot be extended to semi-infinite linear inequality systems. Finally, we give some simple sufficient conditions for the existence of a global error bound for semi-infinite linear inequality systems.