On Error Bounds of Inequalities in Asplund Spaces

TL;DR

This paper investigates error bounds for inequalities in Asplund spaces, extending dual characterization results from convex to non-convex cases using subdifferentials, with specific results for composite-convex functions.

Contribution

It extends dual characterization of error bounds from convex to non-convex inequalities in Asplund spaces, including necessary and sufficient conditions for composite-convex functions.

Findings

Necessary dual conditions in terms of subdifferentials for general inequalities.

Sufficient dual conditions for composite-convex inequalities.

Extension of convex inequality results to non-convex cases.

Abstract

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 of 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. Even though the results of the dual characterization on the error bounds of a general inequality (if one drops the convexity assumption) may not be valid, several necessary dual conditions are still obtained in terms of Fr\'echet/Mordukhovich subdifferentials of the concerned function at points in the solution set. Moreover, for an inequality defined by a composite-convex 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 results on dual characterizations of convex inequalities to the possibly non-convex case.