Characterizations of Super-regularity and its Variants

TL;DR

This paper introduces Clarke super-regularity, a property linking super-regular sets to subsmoothness, and explores its implications for the regularity of functions and their epigraphs, filling a gap in nonconvex optimization theory.

Contribution

It establishes the equivalence between Clarke super-regularity and subsmoothness, connecting set regularity to function regularity and extending the theory of nonconvex optimization.

Findings

Clarke super-regularity is equivalent to subsmoothness.

Approximately convex functions have Clarke super-regular epigraphs.

The paper discusses new classes of regularity based on epigraph properties.

Abstract

Convergence of projection-based methods for nonconvex set feasibility problems has been established for sets with ever weaker regularity assumptions. What has not kept pace with these developments is analogous results for convergence of optimization problems with correspondingly weak assumptions on the value functions. Indeed, one of the earliest classes of nonconvex sets for which convergence results were obtainable, the class of so-called super-regular sets introduced by Lewis, Luke and Malick (2009), has no functional counterpart. In this work, we amend this gap in the theory by establishing the equivalence between a property slightly stronger than super-regularity, which we call Clarke super-regularity, and subsmootheness of sets as introduced by Aussel, Daniilidis and Thibault (2004). The bridge to functions shows that approximately convex functions studied by Ngai, Luc and Th\'era (2000) are those which have Clarke super-regular epigraphs. Further classes of regularity of functions based on the corresponding regularity of their epigraph are also discussed.