Asymptotic constraint qualifications and global error bounds for convex inequalities

In this paper we study various asymptotic constraint qualifications for the existence of global error bounds for approximate solutions of convex inequalities. Many known conditions that ensure the existence of such a global error bound are shown to be equivalent to one of the following three conditions: (i) the bounded excess condition, (ii) Slater condition together with the asymptotic constraint qualification defined by Auslender and Crouzeix, and (iii) positivity of normal directional derivatives of the maximum of the constraint functions introduced by Lewis and Pang.