Abstract

Ordinary distance functions, oriented distance functions and farthest distance functions are characterized in terms of convex conjugates and Fenchel subdifferentials.

Author information

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

Juan Enrique Martínez-Legaz

(1) Department d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona, Barcelona, Spain
(2) Barcelona Graduate School of Mathematics, Barcelona, Spain

juanenrique.martinez.legaz@uab.cat

Keywords

  • Distance function
  • oriented distance function
  • farthest distance function
  • Fenchel conjugate
  • Fenchel subdifferential

Mathematics Subject Classification

46B20, 26B25

References

  1. M. V. Balashov, G. E. Ivanov: On farthest points of sets, Math. Notes 80 (2006) 159–166; translation from Mat. Zametki 80 (2006) 63–170.https://doi.org/10.1007/s11006-006-0123-6Google Scholar
  2. J. V. Burke, M. C. Ferris, M. Qian: On the Clarke subdifferential of the distance function of a closed set, J. Math. Anal. Appl. 166 (1992) 199–213.https://doi.org/10.1016/0022-247X(92)90336-CGoogle Scholar
  3. A. Coulibaly, J.-P. Crouzeix: Condition numbers and error bounds in convex programming, Math. Programming 116 (2009) 79–113.https://doi.org/10.1007/s10107-007-0132-6Google Scholar
  4. S. Fitzpatrick: Metric projections and the differentiability of distance functions, Bull. Aust. Math. Soc. 22 (1980) 291–312.https://doi.org/10.1017/S0004972700006596Google Scholar
  5. J. B. Hiriart-Urruty: Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Oper. Res. 4 (1979) 79–97.https://doi.org/10.1287/moor.4.1.79Google Scholar
  6. J. B. Hiriart-Urruty: New concepts in nondifferentiable programming, Bull. Soc. Math. France, Mémoire 60 (1979) 57–85.https://doi.org/10.24033/msmf.261Google Scholar
  7. J. B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms. I: Fundamentals, Springer, Berlin (1993).https://doi.org/10.1007/978-3-662-02796-7Google Scholar
  8. A. D. Ioffe: Variational Analysis of Regular Mappings. Theory and Applications, Springer, Cham (2017).https://doi.org/10.1007/978-3-319-64277-2Google Scholar
  9. G. E. Ivanov: Farthest points and strong convexity of sets, Math. Notes 87 (2010) 355–366; translation from Mat. Zametki 87 (2010) 382–395.https://doi.org/10.1134/S0001434610030065https://doi.org/10.4213/mzm4428Google Scholar
  10. H. Luo, X. Wang, B. Lukens: Variational analysis on the signed distance functions, J. Optim. Theory Appl. 180 (2019) 751–774.https://doi.org/10.1007/s10957-018-1414-2Google Scholar
  11. B. S. Mordukhovich: Variational Analysis and Generalized Differentiation I, Springer, New York (2006).https://doi.org/10.1007/3-540-31247-1Google Scholar
  12. F. Nacry, V. A. T. Nguyen, L. Thibault: Farthest distance function to strongly convex sets, J. Convex Analysis 30 (2023) 1217–1240.https://doi.org/10.68381/jca30055Google Scholar
  13. J.-P. Penot: Calculus without Derivatives, Springer, New York (2013).https://doi.org/10.1007/978-1-4614-4538-8Google Scholar
  14. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  15. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar

Suggested citation

J. E. Martínez-Legaz. “Dual Characterizations of Three Distance Functions.” Journal of Convex Analysis 31 (2024), No. 4, 1203–1212. https://doi.org/10.68381/jca31059

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.