On Error Bounds of Inequalities in Asplund Spaces

Error bounds are central objects in optimization theory and its applications. They were for a long time restricted only to the theory before becoming over the course of time a field by itself. This paper is devoted to the study of error bounds of a general inequality defined by a proper lower semicontinuous function on an Asplund space. If one drops the convexity assumption, the dual characterization of error bounds for a general inequality may not be valid, but even in this case, several dual necessary conditions are still obtained in terms of Fréchet/Mordukhovich subdifferentials of the concerned function at points in the solution set. Moreover, for an inequality defined by a convex-composite function that is to say by a function which is the composition of a convex function with a smooth mapping, such dual conditions also turn out to be sufficient to have the error bound property. Our work is an extension of the results on dual characterizations of convex inequalities to the possibly non-convex case.