On Global Error Bounds for Convex Inequalities Systems

In this paper, we first present necessary and sufficient conditions for the existence of global error bounds for a convex function without additional conditions on the function or the solution set. In particular, we obtain characterizations of such global error bounds in Euclidean spaces, which are often simple to check. Second, we prove that under a suitable assumption the subdifferential of the supremum function of an arbitrary family of convex continuous functions coincides with the convex hull of the subdifferentials of functions corresponding to the active indices at given points. As applications, we study the existence of global error bounds for infinite systems of linear and convex inequalities. Several examples are provided as well to explain the advantages of our results with existing ones in the literature.