Abstract

We study the properties of the normal cone to a proximally smooth set. We give a complete characterization of a proximally smooth set through the monotonicity properties of its normal cone in an arbitrary uniformly convex and uniformly smooth Banach space. We also give the exact bounds for the right-hand side in the monotonicity inequality for the normal cone in terms of the moduli of smoothness and convexity of a Banach space.

Author information

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

Grigory M. Ivanov

DCG, FSB, École Polytechnique de Lausanne, Route Cantonale, 1015 Lausanne, Switzerland

grimivanov@gmail.com

Keywords

  • Proximal smoothness
  • hypomonotonicity
  • variational inequality
  • Fréchet normal cone

Mathematics Subject Classification

52A05, 52A40

References

  1. F. H. Clarke, R. J. Stern, P. R. Wolenski: Proximal smoothness and lower–C 2 property, J. Convex Analysis 2(1) (1995) 117–144.https://doi.org/10.68381/jca02008Google Scholar
  2. R. A. Poliquin, R. T. Rockafellar: Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 368 (1996) 1805–1838.https://doi.org/10.1090/S0002-9947-96-01544-9Google Scholar
  3. 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
  4. J.-P. Vial: Strong and weak convexity of sets and functions, Math. Oper. Res. 8(2) (1983) 231–259.https://doi.org/10.1287/moor.8.2.231Google Scholar
  5. G. E. Ivanov: Weakly Convex Sets and Functions. Theory and Applications, Fizmatlit (in Russian) (2006).Google Scholar
  6. 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
  7. 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
  8. G. Colombo, L. Thibault: Prox-regular sets and applications, in: Handbook of Nonconvex Analysis, Int. Press Somerville (2010) 99–182.Google Scholar
  9. G. E. Ivanov, G. M. Ivanov: Relationship between supporting conditions in Banach spaces, Proceedings of MIPT 3(1) (2011) 74–80.Google Scholar
  10. M. V. Balashov, G. E. Ivanov: Weakly convex and proximally smooth sets in Banach spaces, Izv. RAN. Ser. Mat. 73(3) (2009) 23–66.https://doi.org/10.4213/im2646Google Scholar
  11. B. S. Mordukhovich: Penalty functions and necessary conditions for an extremum in non-smooth and non-convex optimization problems, Russian Mathematical Surveys 36(1) (1981) 242–243.https://doi.org/10.1070/RM1981v036n01ABEH002553Google Scholar
  12. R. T. Rockafellar: Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981) 424–436.https://doi.org/10.1287/moor.6.3.424Google Scholar
  13. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces II, Springer-Verlag, Berlin (1979).https://doi.org/10.1007/978-3-662-35347-9Google Scholar
  14. A. J. Guirao, M. Ivanov, S. Lajara: On moduli of smoothness and squareness, J. Convex Analysis 17 (2010) 441–449.https://doi.org/10.68381/jca17031Google Scholar
  15. M. I. Kadets, B. S. Mityagin: Complemented subspaces in Banach spaces, Russian Mathematical Surveys 28(6) (1973) 77.https://doi.org/10.1070/RM1973v028n06ABEH001630Google Scholar
  16. M. I. Kadets, M. G. Snobar: Some functionals over a compact Minkovskii space, Mathematical Notes of the Academy of Sciences of the USSR 10(4) (1971) 694–696.https://doi.org/10.1007/BF01106467Google Scholar
  17. P. M. Gruber: Convex and Discrete Geometry, Springer-Verlag, Berlin-Heidelberg (2007).Google Scholar
  18. T. Figiel: On the moduli of convexity and smoothness, Studia Math 56(2) (1976) 121–155.https://doi.org/10.4064/sm-56-2-121-155Google Scholar
  19. J. M. Borwein, J. D. Vanderwerff: Convex Functions: Constructions, Characterizations and Counterexamples, Encyclopedia of Mathematics and its Applications, Cambridge University Press (2010).https://doi.org/10.1017/CBO9781139087322Google Scholar
  20. D. Amir: Characterizations of Inner Product Spaces, Birkhäuser Verlag, Basel (1986).https://doi.org/10.1007/978-3-0348-5487-0Google Scholar
  21. J. Diestel: Geometry of Banach Spaces - Selected Topics, Lecture Notes in Mathematics 485, Springer-Verlag, Berlin-Heidelberg (1975).https://doi.org/10.1007/BFb0082079Google Scholar
  22. G. M. Ivanov: Modulus of supporting convexity and supporting smoothness, Eurasian Math. J. 6(1) (2015) 26–40.Google Scholar
  23. S. V. Konyagin: On approximative properties of arbitrary closed sets in Banach spaces (in Russian), Fund. Prikl. Mat. 3(4) (1997) 979–989.Google Scholar

Suggested citation

G. M. Ivanov. “Hypomonotonicity of the Normal Cone and Proximal Smoothness.” Journal of Convex Analysis 24 (2017), No. 4, 1313–1339. https://doi.org/10.68381/jca24076

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.