Abstract

We provide a characterization of uniformly lower regular functions defined on a Hilbert space. To this end we introduce and study a property we call epi prox-regularity of an epigraph set which slightly differs from the well-known prox-regularity property of a set.

Author information

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

Keywords

  • Prox-regular set
  • uniformly prox-regular set
  • primal lower nice function
  • proximal normal
  • distance function
  • metric projection mapping
  • Hilbert space

Mathematics Subject Classification

49J52, 49J53

References

  1. S. Adly, F. Nacry, L. Thibault: Preservation of prox-regularity of sets with applications to constrained optimization, SIAM J. Optim. 26 (2016) 448–473.https://doi.org/10.1137/15M1032739Google Scholar
  2. S. Adly, F. Nacry, L. Thibault: Discontinuous sweeping process with prox-regular sets, ESAIM, Control Optim. Calc. Var. 23/4 (2017) 1293–1329.https://doi.org/10.1051/cocv/2016053Google Scholar
  3. S. Adly, F. Nacry, L. Thibault: Prox-regularity approach to generalized equations and image projection, ESAIM, Control Optim. Calc. Var. 24 (2018) 677–708.https://doi.org/10.1051/cocv/2017052Google Scholar
  4. S. Adly, F. Nacry, L. Thibault: New metric properties for prox-regular sets, Math. Programming 189/1 (2021) 7–36.https://doi.org/10.1007/s10107-020-01525-xGoogle Scholar
  5. S. Adly, F. Nacry, L. Thibault: Prox-regular sets and Legendre-Fenchel transform related to separation properties, Optimization 71/7 (2022) 2097–2129.https://doi.org/10.1080/02331934.2020.1852404Google Scholar
  6. M. V. Balashov, G. E. Ivanov: Weakly convex and proximally smooth sets in Banach spaces, Izv. Math. 73 (2009) 455–499.https://doi.org/10.1070/IM2009v073n03ABEH002454Google Scholar
  7. F. Bernard, L. Thibault: Prox-regular functions and sets in Banach spaces, Set-Valued Analysis 12 (2004) 25–47.https://doi.org/10.1023/B:SVAN.0000023403.87092.a2Google Scholar
  8. F. Bernard, L. Thibault, D. Zagrodny: Integration of primal lower nice functions in Hilbert spaces, J. Optim. Theory Appl. 124/3 (2005) 561–579.https://doi.org/10.1007/s10957-004-1174-zGoogle Scholar
  9. F. Bernard, L. Thibault, N. Zlateva: Characterizations of prox-regular sets in uniformly convex Banach spaces, J. Convex Analysis 13 (2006) 525–559.https://doi.org/10.68381/jca13044Google Scholar
  10. F. Bernard, L. Thibault, N. Zlateva: Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties, Trans. Amer. Math. Soc. 363 (2011) 2211–2247.https://doi.org/10.1090/S0002-9947-2010-05261-4Google Scholar
  11. J. M. Borwein, J. R. Giles: The proxima normal formula in Banach space, Trans. Amer. Math. Soc. 302 (1987) 371–381.https://doi.org/10.1090/S0002-9947-1987-0887515-5Google Scholar
  12. J. M. Borwein, H. M. Strójwas: Proximal analysis and boundaries of closed sets in Banach space. I: Theory, Canad. J. Math. 38/2 (1986) 428–472.https://doi.org/10.4153/CJM-1987-019-4Google Scholar
  13. 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
  14. G. Colombo, L. Thibault: Prox-regular sets and applications, in: Handbook of Nonconvex Analysis and Applications, D. Y. Gao et al. (eds.), International Press, Somerville (2010) 99–182.Google Scholar
  15. C. Combari, A. Elhilali Alaoui, A. Levy, R. A. Poliquin, L. Thibault: Convex composite functions in Banach spaces and the primal-lower-nice property, Proc. Amer. Math. Soc. 126 (1998) 3701–3708.https://doi.org/10.1090/S0002-9939-98-04324-XGoogle Scholar
  16. R. Correa, A. Jofré, L. Thibault: Characterization of lower semicontinuoud convex fuctions, Proc. Amer. Math. Soc. 116 (1992) 61–72.https://doi.org/10.1090/S0002-9939-1992-1126193-4Google Scholar
  17. H. Federer: Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418–491.https://doi.org/10.1090/S0002-9947-1959-0110078-1Google Scholar
  18. V. Goncharov, G. E. Ivanov: Strong and weak convexity of closed sets in a Hilbert space, in: Operations Research, Engineering, and Cyber Security, Nea Peramos 2015, N. J. Daras et al. (eds.), Springer Optimization and its Applications Series 113, Springer, Cham (2017) 259–297.https://doi.org/10.1007/978-3-319-51500-7_12Google Scholar
  19. A. D. Ioffe: Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990) 175–192.https://doi.org/10.1112/jlms/s2-41.1.175Google Scholar
  20. G. E. Ivanov: Weak convexity in the sense of Vial and Efimov-Stechkin (Russian), Izv. Ross. Akad. Nauk Ser. Mat. 69/6 (2005), 35–60; English translation in Izv. Math. 69/6 (2005) 1113–1135.https://doi.org/10.4213/im665https://doi.org/10.1070/IM2005v069n06ABEH002292Google Scholar
  21. M. Ivanov, N. Zlateva: On primal lower-nice property, C. R. Acad. Bulg. Sci. 54/11 (2001) 5–10.Google Scholar
  22. M. Ivanov, N. Zlateva: Subdifferential characterization of primal lower-nice functions on smooth Banach spaces, C. R. Acad. Bulg. Sci. 57/5 (2004) 13–18.Google Scholar
  23. I. Kecis, 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
  24. I. Kecis, L. Thibault: Moreau envelopes of s-lower regular functions, Nonlinear Analysis 127 (2015) 157–181.https://doi.org/10.1016/j.na.2015.06.015Google Scholar
  25. M. Konstantinov, N. Zlateva: Direct proofs of intrinsic properties of prox-regular sets in Hilbert spaces, J. Appl. Analysis 29/1 (2023) 143–149.https://doi.org/10.1515/jaa-2022-1010Google Scholar
  26. A. Levy, R. A. Poliquin, L. Thibault: Partial extension of Attouch’s theorem with applications to proto-derivatives of subgradient mappings, Trans. Amer. Math. Soc. 347 (1995) 1269–1294.https://doi.org/10.1090/S0002-9947-1995-1290725-3Google Scholar
  27. 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 & Statistics 50, Springer, New York (2013) 521–553.https://doi.org/10.1007/978-1-4614-7621-4_25Google Scholar
  28. F. Nacry, L. Thibault: Regularization of sweeping process: old and new, Pure Appl. Funct. Analysis 4 (2019) 59–117.Google Scholar
  29. R. A. Poliquin: Integration of subdifferentials of nonconvex functions, Nonlinear Analysis Theor. Meth. Appl. 17 (1991) 385–398.https://doi.org/10.1016/0362-546X(91)90078-FGoogle Scholar
  30. R. A. Poliquin: An extension of Attouch’s Theorem and its application to second-order epidifferentiation of convexly composite functions, Trans. Amer. Math. Soc. 332 (1992) 861–874.https://doi.org/10.1090/S0002-9947-1992-1145732-5Google Scholar
  31. R. A. Poliquin, R. T. Rockafellar: Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 348 (1996) 1805–1838.https://doi.org/10.1090/S0002-9947-96-01544-9Google Scholar
  32. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352/11 (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  33. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der Mathematischen Wissenschaften 317 Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  34. L. Thibault: Unilateral Variational Analysis in Banach Spaces. I: General Theory, II: Special Classes of Functions and Sets, World Scientific, Singapore (2023).https://doi.org/10.1142/12797Google Scholar
  35. L. Thibault, D. Zagrodny: Integration of subdifferentials of lower semicontinuous functions, J. Math. Analysis Appl. 189 (1995) 33–58.https://doi.org/10.1006/jmaa.1995.1003Google Scholar
  36. J.-P. Vial: Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (1983) 231–259.https://doi.org/10.1287/moor.8.2.231Google Scholar

Suggested citation

M. Konstantinov, N. Zlateva. “Epigraphical Characterization of Uniformly Lower Regular Functions in Hilbert Spaces.” Journal of Convex Analysis 30 (2023), No. 4, 1139–1153. https://doi.org/10.68381/jca30051

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.