Characterizations of Super-Regularity and Its Variants

Aris Danillidis (), D. Russell Luke () and Matthew Tam ()
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Aris Danillidis: Universidad de Chile, DIM-CMM
D. Russell Luke: Universität Göttingen, Inst. Numerische & Angewandte Mathematik
Matthew Tam: University of Göttingen

Chapter Chapter 6 in Splitting Algorithms, Modern Operator Theory, and Applications, 2019, pp 137-152 from Springer

Abstract: 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 (Lewis et al., Comput. Math. 9(4), 485–513, 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 (Amer. Math. Soc. 357, 1275–1301, 2004). The bridge to functions shows that approximately convex functions studied by Ngai, Luc and Thera (J. Nonlinear Convex Anal. 1, 155–176, 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.

Keywords: Super-regularity; Subsmoothness; Approximately convex; 49J53; 26B25; 49J52; 65K10 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/978-3-030-25939-6_6

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