Abstract

We consider a supporting condition for a convex compact set in Rn\mathbb{R}^n. This condition is typical and it is closely related to the local Lipschitz differentiability of the supporting function. With the help of the considered condition we prove a local Lipschitz condition for supporting elements and for nearest and farthest points of a convex compact set. Some applications for numerical algorithms are discussed.

Author information

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

Maxim V. Balashov

Trapeznikov Institute of Control Sciences, Moscow, Russia

balashov73@mail.ru

Keywords

  • Supporting function
  • subdifferential
  • supporting condition
  • nonsmooth analysis

Mathematics Subject Classification

49J52, 49J53, 52A20; 47H09

References

  1. M. V. Balashov: Antidistance and antiprojection in the Hilbert space, J. Convex Analysis 22/2 (2015) 521–536.https://doi.org/10.68381/jca22024Google Scholar
  2. M. V. Balashov: About the gradient projection algorithm for a strongly convex function and a proximally smooth set, J. Convex Analysis 24/2 (2017) 493–500.https://doi.org/10.68381/jca24031Google Scholar
  3. M. V. Balashov: Sufficient conditions for the linear convergence of an algorithm for finding the metric projection of a point onto a convex compact set, Math. Notes 113/5 (2023) 632–641.https://doi.org/10.1134/S0001434623050036Google Scholar
  4. M. V. Balashov, K. Z. Biglov: The strong convexity supporting condition and the Lipschitz condition for the metric projection, Math. Notes 115/2 (2024) 164–172.https://doi.org/10.1134/S0001434624010152Google Scholar
  5. M. V. Balashov, K. Z. Biglov, A. A. Tremba: On some problems with multivalued mappings, Automation Remote Control 85/5 (2024) 491–511.https://doi.org/10.31857/S0005117924050047Google Scholar
  6. M. V. Balashov, M. O. Golubev: About the Lipschitz property of the metric projection in the Hilbert space, J. Math. Analysis Appl. 394/2 (2012) 545–551.https://doi.org/10.1016/j.jmaa.2012.05.024Google Scholar
  7. M. V. Balashov, G. E. Ivanov: On farthest points of sets, Math. Notes 80/2 (2006) 159–166.https://doi.org/10.1007/s11006-006-0123-6Google Scholar
  8. V. I. Bogachev: Measure Theory Vol. 1, Springer, Berlin (2007).https://doi.org/10.1007/978-3-540-34514-5_1Google Scholar
  9. F. H. Clarke, J. R. Stern, P. R. Wolenski: Proximal smoothness and the lower-C 2 property, J. Convex Analysis 2/1-2 (1995) 117–144.https://doi.org/10.68381/jca02008Google Scholar
  10. H. Federer: Curvature measures, Trans. Amer. Math. Soc. 93/3 (1959) 418–491.https://doi.org/10.1090/S0002-9947-1959-0110078-1Google Scholar
  11. H. Frankowska, C. Olech: R-convexity of the integral of the set-valued functions, Contributions to Analysis and Geometry, Suppl. Amer. J. Math. (1981) 117–129.Google Scholar
  12. V. V. Goncharov, G. E. Ivanov: Strong and weak convexity of closed sets in a Hilbert space, in: Operations Research, Engineering, and Cyber Security. Trends in Applied Mathematics and Technology, N. J. Daras et al. (eds.), Nea Peramos, Greece, May 2015, Springer Optimization and Its Applications Vol. 113, Springer, Cham (2017) 259–297.https://doi.org/10.1007/978-3-319-51500-7_12Google Scholar
  13. G. E. Ivanov: Extremal problems for the distance between elements of two sets, J. Convex Analysis 29/3 (2022) 767–788.https://doi.org/10.68381/jca29044Google Scholar
  14. G. E. Ivanov, E. S. Polovinkin: On strongly convex differential games, Diff. Equations 31/10 (1995) 1603–1612.Google Scholar
  15. S. Lojaciewicz: Some properties of accessible sets in nonlinear control systems, Ann. Polon. Math. 36 (1979) 123–137.https://doi.org/10.4064/ap-36-2-123-137Google Scholar
  16. A. E. Mayer: Der Inhalt der Gleichdicke, Math. Ann. 110 (1935) 97–127.https://doi.org/10.1007/BF01448020Google Scholar
  17. A. E. Mayer: Eine Überkonvexität, Math. Zeitschrift 39 (1935) 511–531.https://doi.org/10.1007/BF01201371Google Scholar
  18. F. Nacry, N. V. A. Thuong, L. Thibault: Farthest distance function to strongly convex sets, J. Convex Analysis 30/4 (2023) 1217–1240.https://doi.org/10.68381/jca30055Google Scholar
  19. I. Necoara, Yu. Nesterov, F. Glineur: Linear convergence of first order methods for non-strongly convex optimization, Math. Programming Ser. A 175/1-2 (2019) 69–107.https://doi.org/10.1007/s10107-018-1232-1Google Scholar
  20. A. Pliś: Uniqueness of optimal trajectories for non-linear control systems, Ann. Polon. Math. 29 (1975) 397–401.https://doi.org/10.4064/ap-29-4-397-401Google Scholar
  21. E. S. Polovinkin: Strongly convex analysis, Sb. Math. 187/2 (1996) 259–286.https://doi.org/10.1070/SM1996v187n02ABEH000111Google Scholar
  22. Yu. G. Reshetnyak: On a generalization of convex surfaces, Mat. Sb. (N.S.) 40(82)/3 (1956) 381–398.Google Scholar
  23. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  24. L. Thibault: Jean Jacques Moreau. A selected review of his mathematical works, J. Convex Analysis 23/1 (2016) 005–022.https://doi.org/10.68381/jca23001Google Scholar
  25. V. M. Veliov: On the convexity of integrals of multivalued mappings: application in control theory, J. Optimization Theory Appl. 54/3 (1987) 541–563.https://doi.org/10.1007/BF00940201Google Scholar
  26. J.-Ph. 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

Suggested citation

M. V. Balashov, K. Z. Biglov. “Lipschitz Differentiability of the Supporting Function from a Geometric Viewpoint, the Nearest and the Farthest Points.” Journal of Convex Analysis 32 (2025), No. 1, 107–118. https://doi.org/10.68381/jca32008

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.