Abstract

A Lipschitz-like property of subdifferential mappings of convex continuous functions on Hilbert spaces is investigated. It is shown that the subdifferential mapping of such a function admits a Lipschitz-like property at all points from a dense subset of its domain. In particular, Lipschitz-like properties of subdifferentials of distance functions from sets with convex complements are discovered.

Author information

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

Keywords

  • Convex functions
  • Aubin Lipschitz-like properties
  • best approximation
  • metric projection mapping
  • Borel sets

Mathematics Subject Classification

52A41; 49J50

References

  1. J.-P. Aubin: Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984) 87–111.https://doi.org/10.1287/moor.9.1.87Google Scholar
  2. V. S. Balaganskii: On approximation properties of sets with convex complement, Math. Notes 57 (1995) 15–20.https://doi.org/10.1007/BF02309389Google Scholar
  3. V. Barbu, T. Precupanu: Convexity and Optimization in Banach Spaces, 4th ed., Springer, Berlin (2012).https://doi.org/10.1007/978-94-007-2247-7Google Scholar
  4. H. H. Bauschke, P. L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, Berlin (2011).https://doi.org/10.1007/978-1-4419-9467-7Google Scholar
  5. J. M. Borwein: Future challenges for variational analysis, analysis and generalized differentiation in optimization and control, in: Springer Optimization and its Applications 47, Springer, Cham (2010) 95–105.https://doi.org/10.1007/978-1-4419-0437-9_4Google Scholar
  6. J. M. Borwein, J. D. Vanderwerff: Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, Cambridge (2010).https://doi.org/10.1017/CBO9781139087322Google Scholar
  7. A. Cabot, A. Jourani, L. Thibault, D. Zagrodny: The attainment set of ϕ-envelope and genericity properties, Mathematica Scandinavica 124 (2019) 203–246.https://doi.org/10.7146/math.scand.a-110766Google Scholar
  8. S. Fitzpatrick: Metric projections and the differentiability of distance functions, Bull. Austral. Math. Soc. 22 (1980) 291–312.https://doi.org/10.1017/S0004972700006596Google Scholar
  9. R. J. Gardner, W. F. Pfeffer: Borel Measures, in: Handbook of Set-Theoretic Topology, Chapter 22, K. Kunen, J. E. Vaughan (eds.), North-Holland, Amsterdam (1984).https://doi.org/10.1016/B978-0-444-86580-9.50025-2Google Scholar
  10. J.-B. Hiriart-Urruty: New concepts in nondifferentiable programming, Mém. Soc. Math. France 60 (1979) 57–85.https://doi.org/10.24033/msmf.261Google Scholar
  11. A. D. Ioffe, V. M. Tikhomirov: Theory of Extremal Problems, North-Holland, Amsterdam (1979).Google Scholar
  12. A. Jourani, L. Thibault, D. Zagrodny: C 1,ω(·) -regularity and Lipschitz-like properties of subdifferential, Proc. London Math. Soc. (3) 105 (2012) 189–223.https://doi.org/10.1112/plms/pdr062Google Scholar
  13. A. Jourani, L. Thibault, D. Zagrodny: Differential properties of the Moreau envelope, J. Func. Analysis 266 (2014) 1185–1237.https://doi.org/10.1016/j.jfa.2013.11.008Google Scholar
  14. A. Jourani, L. Thibault, D. Zagrodny: The NSLUC property and Klee envelope, Math. Ann. V. 365 I. 3-4 (2016) 923–967.https://doi.org/10.1007/s00208-015-1283-zGoogle Scholar
  15. K.-S. Lau: On almost Chebyshev subsets in reflexive Banach spaces, Indiana Univ. Math. J. 27 (1978) 791–795.https://doi.org/10.1512/iumj.1978.27.27051Google Scholar
  16. J. C. Oxtoby: Measure and Category. A Survey of the Analogies between Topological and Measure Spaces, 2nd ed., Springer, New York(1980).https://doi.org/10.1007/978-1-4684-9339-9_22Google Scholar
  17. K. R. Parthasarathy: Probability Measures on Metric Spaces, Academic Press, New York (1967).https://doi.org/10.1016/B978-1-4832-0022-4.50007-7Google Scholar
  18. D. A. Salamon: Measure and Integration, European Mathematical Society, Zurich (2016).https://doi.org/10.4171/159Google Scholar
  19. R. L. Wheeden, A. Zygmund: Measure and Integral. An Introduction to Real Analysis, Marcel Dekker, New York et al. (1977).https://doi.org/10.1201/b15702Google Scholar
  20. K. Yosida: Functional Analysis, 6th ed., Springer, Berlin (1980).Google Scholar
  21. D. Zagrodny: Convergences of subgradients of sequences of convex functions, Nonlinear Analysis 84 (2013) 84–90.https://doi.org/10.1016/j.na.2013.01.015Google Scholar
  22. D. Zagrodny: On closures of preimages of metric projection mappings in Hilbert spaces, Set-Valued Var. Analysis 23 (2015) 581–612.https://doi.org/10.1007/s11228-015-0350-7Google Scholar
  23. D. Zagrodny: On the strong convergence of subgradients of convex functions, J. Optim. Theory Appl. 178(2) (2018) 411–423.https://doi.org/10.1007/s10957-018-1276-7Google Scholar
  24. D. Zagrodny: The strong convergence of subgradients of convex functions along directions: Perspectives and open problems, J. Optim. Theory Appl. 178(2) (2018) 660–671.https://doi.org/10.1007/s10957-018-1323-4Google Scholar
  25. N. V. Zhivkov: Metric projections and antiprojections in strictly convex normed spaces, C. R. Acad. Bulgar. Sci. 31(4) (1978) 369–372.Google Scholar

Suggested citation

D. Zagrodny. “Lipschitz-like Property of Subdifferential Mapping of Convex Continuous Functions on Hilbert Space.” Journal of Convex Analysis 27 (2020), No. 2, 753–776. https://doi.org/10.68381/jca27038

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.