Abstract

We consider a class of subsets of a Banach space whose normal cones satisfy a kind of uniformity property. Here the normal cone is associated with a general subdifferential as defined in an appendix for the sake of clarity. We examine to what extent this property depends on the choice of the subdifferential. We relate this property to approximate convexity in the sense of Ngai, Luc and Théra, subsmoothness in the sense of Aussel, Daniilidis, Thibault and super-regularity in the sense of Lewis, Luke and Malick, showing coincidence in appropriate spaces. This class of sets is larger than the classes of amenable subsets and prox-regular subsets. We also explore related properties for functions and multifunctions. We devise some permanence properties under usual operations.

Author information

Contact details are reproduced from the original publication and may be historical.

Jean-Paul Penot

Sorbonne Universities, UPMC Univ Paris 6, CNRS – Lab. J.-L. Lions, Paris, France

penotj@ljll.math.upmc.fr

Keywords

  • Approximate convexity
  • approximate monotonicity
  • error bound
  • normal cone
  • subconvexity
  • subdifferential

Mathematics Subject Classification

49J52, 49J53, 52A01, 52A30, 58C06, 90C26

References

  1. D. Aussel, J.-N. Corvelec, M. Lassonde: Mean value property and subdifferential criteria for lower semicontinuous functions, Trans. Amer. Math. Soc. 347/10 (1995) 4147–4161.https://doi.org/10.1090/S0002-9947-1995-1307998-0Google Scholar
  2. D. Aussel, A. Daniilidis, L. Thibault: Subsmooth sets: functional characterizations and related concepts, Trans. Amer. Math. Soc. 357/4 (2005) 1275–1301.https://doi.org/10.1090/S0002-9947-04-03718-3Google Scholar
  3. D. Azé: Inner conditions for error bounds and metric subregularity of multifunctions, http://www.optimization-online.org/DBHTML/2017/09/6209.html, corrected in: Characterizations for existence of multipliers in mathematical programming in Banach spaces, https://www.researchgate.net/publication/341113349 (2020).Google Scholar
  4. F. Bernard, L. Thibault: Prox-regularity of functions and sets in Banach spaces, Set-Valued Analysis 12 (2004) 25–47.https://doi.org/10.1023/B:SVAN.0000023403.87092.a2Google Scholar
  5. J. M. Borwein, Q. J. Zhu: Techniques of Variational Analysis, CMS Books in Mathematics, Canadian Mathematical Society, Springer, New York (2005).Google Scholar
  6. M. Bounkhel, L. Thibault: On various notions of regularity of sets in non-smooth analysis, Nonlin. Analysis Theory Meth. Appl. 43 (2002) 223–246.https://doi.org/10.1016/S0362-546X(00)00183-8Google Scholar
  7. F. H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics, Philadelphia (1990).https://doi.org/10.1137/1.9781611971309Google Scholar
  8. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Springer, New York (1998).Google Scholar
  9. F. H. Clarke, R. J. Stern, P. R. Wolenski: Proximal smoothness and the lower C 2 property, J. Convex Analysis 2 (1995) 117–144.https://doi.org/10.68381/jca02008Google Scholar
  10. G. Colombo, V. V. Goncharov: Variational inequalities and regularity properties of closed sets in Hilbert spaces, J. Convex Analysis 8/1 (2001) 197–221.https://doi.org/10.68381/jca08009Google Scholar
  11. G. Colombo, L. Thibault: Prox-regular sets and applications, in: Handbook of Nonconvex Analysis and Applications, International Press, Somerville (2010) 99–182.Google Scholar
  12. A. Daniilidis, P. Georgiev: Approximate convexity and submonotonicity, J. Math. Analysis Appl. 291 (2004) 292–301.https://doi.org/10.1016/j.jmaa.2003.11.004Google Scholar
  13. A. Daniilidis, F. Jules, M. Lassonde: Subdifferential characterization of approximate convexity: the lower semicontinuous case, Math. Programming Ser. B 116 (2009) 115–127.https://doi.org/10.1007/s10107-007-0127-3Google Scholar
  14. A. Daniilidis, L. Thibault: Subsmooth sets and metrically subsmooth sets and functions in Banach space, unpublished manuscript, University of Montpellier 2 (2008).Google Scholar
  15. A. D. Ioffe: Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984) 389–416.https://doi.org/10.1090/S0002-9947-1984-0719677-1Google Scholar
  16. A. D. Ioffe: Codirectional compactness, metric regularity and subdifferential calculus, in: Constructive, Experimental and Nonlinear Analysis, M. Théra (ed.), Canadian Math. Soc. Conf. Proc. 27, American Math. Society, Providence (1999) 123–153.Google Scholar
  17. A. D. Ioffe: Variational Analysis of Regular Mappings. Theory and Applications, Springer, Berlin (2017).https://doi.org/10.1007/978-3-319-64277-2Google Scholar
  18. A. D. Ioffe, J.-P. Penot: Subdifferentials of performance functions and calculus of set-valued mappings, Serdica Math. J. 22 (1996) 359–384.Google Scholar
  19. I. Kessis, L. Thibault: Subdifferential characterization of s-lower regular functions, Appl. Analysis 94/1 (2015) 85–98.https://doi.org/10.1080/00036811.2013.878970Google Scholar
  20. A. S. Lewis, D. R. Luke, J. Malick: Local linear convergence for alternating and averaged nonconvex projections, Found. Comput. Math. 9 (2009) 485–513.https://doi.org/10.1007/s10208-008-9036-yGoogle Scholar
  21. M. Mazade, L. Thibault: Primal lower-nice functions and their Moreau envelopes, in: Computational and Analytical Mathematics, D. H. Bailey et al. (eds.), Springer Proceedings in Mathematics and Statistics 50, Springer, New York (2013) 521–553.https://doi.org/10.1007/978-1-4614-7621-4_25Google Scholar
  22. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation I, Springer, New York (2006).https://doi.org/10.1007/3-540-31247-1Google Scholar
  23. H. V. Ngai, D. T. Luc, M. Théra: Approximate convex functions, J. Nonlin. Convex Analysis 1 (2000) 155–176.Google Scholar
  24. H. V. Ngai, J.-P. Penot: Approximately convex functions and approximately monotone operators, Nonlin. Analysis 66 (2007) 547–564.https://doi.org/10.1016/j.na.2005.11.045Google Scholar
  25. H. V. Ngai, J.-P. Penot: Approximately convex sets, J. Nonlin. Convex Analysis 8/3 (2007) 337–371.Google Scholar
  26. J.-P. Penot: Regularity conditions in mathematical programming, Math. Programming Study 19 (1982) 167–199.https://doi.org/10.1007/BFb0120988Google Scholar
  27. J.-P. Penot: Favorable classes of mappings and multimappings in nonlinear analysis and optimization, J. Convex Analysis 3/1 (1996) 97–116.https://doi.org/10.68381/jca03008Google Scholar
  28. J.-P. Penot: Metric estimates for the calculus of multimappings, Set-Valued Analysis 5 (1997) 291–308.https://doi.org/10.1023/A:1008625212506Google Scholar
  29. J.-P. Penot: The compatibility with order of some subdifferentials, Positivity 6/4 (2002) 413–432.https://doi.org/10.1023/A:1021305308075Google Scholar
  30. J.-P. Penot: Delineating nice classes of nonsmooth functions, Pacific J. Optimization 4/3 (2008) 605–619.Google Scholar
  31. J.-P. Penot: Softness, sleekness and regularity properties in nonsmooth analysis, Nonlin. Analysis 68/9 (2008) 2750–2768.https://doi.org/10.1016/j.na.2007.02.023Google Scholar
  32. J.-P. Penot: Towards a new era in subdifferential analysis?, in: D. H. Bailey et al. (eds.), Computational and Analytical Mathematics, Springer Proceedings in Mathematics and Statistics 50, Springer, New York (2013) 635–665.https://doi.org/10.1007/978-1-4614-7621-4_29Google Scholar
  33. J.-P. Penot: Calculus Without Derivatives, Graduate Texts in Mathematics 266, Springer, New York (2013).https://doi.org/10.1007/978-1-4614-4538-8Google Scholar
  34. J.-P. Penot: Error bounds and multipliers in constrained optimization problems with tolerance, SIAM J. Optimization 29/1 (2019) 522–540.https://doi.org/10.1137/17M1149778Google Scholar
  35. S. Rolewicz: Paraconvex analysis, Control Cybernetics 34/3 (2005) 951–965.Google Scholar
  36. J. E. Spingarn: Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc. 264/1 (1981) 77–89.https://doi.org/10.1090/S0002-9947-1981-0597868-8Google Scholar
  37. J. E. Spingarn: Submonotone mappings and the proximal point algorithm, Numer. Funct. Anal. Opt. 4/2 (1981) 123–150.https://doi.org/10.1080/01630568208816109Google Scholar
  38. L. Thibault: Subsmooth functions and sets, Linear Nonlin. Analysis 4 (2018) 157–269.Google Scholar
  39. L. Thibault: personal communication, May 30th (2020).Google Scholar
  40. L. Thibault, D. Zagrodny: Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Analysis Appl. 189 (1995) 33–58.https://doi.org/10.1006/jmaa.1995.1003Google Scholar
  41. L. Veselý, L. Zajíček: Delta-convex mappings between Banach spaces and applications, Dissertationes Mathematicae 289, Warszawa (1989) 1–48.Google Scholar
  42. X. Y. Zheng, K. F. Ng: Linear regularity for a collection of subsmooth sets in Banach spaces, SIAM J. Optimization 19 (2008) 62–76.https://doi.org/10.1137/060659132Google Scholar
  43. X. Y. Zheng, K. F. Ng: Calmness for a L-subsmooth multifunctions in Banach spaces, SIAM J. Optimization 19 (2009) 1648–1673.https://doi.org/10.1137/080714129Google Scholar

Suggested citation

J.-P. Penot. “Subconvex Functions, Relations and Sets.” Journal of Convex Analysis 29 (2022), No. 4, 1225–1250. https://doi.org/10.68381/jca29067

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.