Journal of Optimization Theory and Applications

, Volume 160, Issue 2, pp 391–414 | Cite as

Directed Subdifferentiable Functions and the Directed Subdifferential Without Delta-Convex Structure

  • Robert Baier
  • Elza Farkhi
  • Vera Roshchina


We show that the directed subdifferential introduced for differences of convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from the directional derivative without using any information on the delta-convex structure of the function. The new definition extends to a more general class of functions, which includes Lipschitz functions definable on o-minimal structure and quasidifferentiable functions.


Nonconvex subdifferentials Directional derivatives Difference of convex (delta-convex, DC) functions Differences of sets 



The research is partially supported by The Hermann Minkowski Center for Geometry at Tel Aviv University, Tel Aviv, Israel and by the University of Ballarat ‘Self-sustaining Regions Research and Innovation Initiative’, an Australian Government Collaborative Research Network (CRN).

We would like to acknowledge the discussions with Jeffrey C.H. Pang starting in October 2009 about Lipschitz definable functions. During the communications he suggested a construction to illustrate the existence of Lipschitz functions which are not quasidifferentiable. This motivated us to consider Example 5.1 which is based on a well-known one-dimensional example of a function that is quasidifferentiable but not a DC function (see [4, Example 3.5]). The authors are also grateful to Aris Daniilidis and Dmitriy Drusvyatskiy for a helpful discussion on Lipschitz definable/semialgebraic functions.


  1. 1.
    Baier, R., Farkhi, E.: The directed subdifferential of DC functions. In: Leizarowitz, A., Mordukhovich, B.S., Shafrir, I., Zaslavski, A.J. (eds.) Nonlinear Analysis and Optimization II: Optimization. A Conference in Celebration of Alex Ioffe’s 70th and Simeon Reich’s 60th Birthdays, Haifa, Israel, 18–24 June, 2008. AMS Contemporary Mathematics, vol. 513, pp. 27–43. AMS and Bar-Ilan University, Providence (2010) CrossRefGoogle Scholar
  2. 2.
    Baier, R., Farkhi, E.: Differences of convex compact sets in the space of directed sets, part I: the space of directed sets. Set-Valued Anal. 9(3), 217–245 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Baier, R., Farkhi, E.: Differences of convex compact sets in the space of directed sets, part II: visualization of directed sets. Set-Valued Anal. 9(3), 247–272 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Baier, R., Farkhi, E., Roshchina, V.: The directed and Rubinov subdifferentials of quasidifferentiable functions. Part I: definition and examples. Nonlinear Anal. 75(3), 1074–1088 (2012). Special Issue on Variational Analysis and Its Applications CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Baier, R., Farkhi, E., Roshchina, V.: The directed and Rubinov subdifferentials of quasidifferentiable functions. Part II: calculus. Nonlinear Anal. 75(3), 1058–1073 (2012). Special Issue on Variational Analysis and Its Applications CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Coste, M.: An introduction to o-minimal geometry. Notes, Institut de Recherche Matématiques de Rennes, 82 pages (November 1999) Google Scholar
  7. 7.
    Daniilidis, A., Pang, J.C.H.: Continuity and differentiability of set-valued maps revisited in the light of tame geometry. J. Lond. Math. Soc. (2) 83(3), 637–658 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Ioffe, A.D.: An invitation to tame optimization. SIAM J. Optim. 19(4), 1894–1917 (2008) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Baier, R., Farkhi, E., Roshchina, V.: On computing the Mordukhovich subdifferential using directed sets in two dimensions. In: Burachik, R., Yao, J.-C. (eds.) Variational Analysis and Generalized Differentiation in Optimization and Control. Springer Optim. Appl., vol. 47, pp. 59–93. Springer, New York (2010) CrossRefGoogle Scholar
  10. 10.
    Rockafellar, R.T.: Convex Analysis, 2nd edn. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1972). First edition (1970) Google Scholar
  11. 11.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Fundamentals. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305. Springer, Berlin (1993) Google Scholar
  12. 12.
    Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Approximation and Optimization, vol. 7. Verlag Peter Lang, Frankfurt/Main (1995). Russian original “Foundations of Nonsmooth Analysis, and Quasidifferential Calculus” published in Nauka, Moscow (1990) zbMATHGoogle Scholar
  13. 13.
    Pallaschke, D., Urbański, R.: Pairs of Compact Convex Sets. Mathematics and Its Applications, vol. 548. Kluwer Academic, Dordrecht (2002) CrossRefzbMATHGoogle Scholar
  14. 14.
    Bolte, J., Daniilidis, A., Lewis, A.: Tame functions are semismooth. Math. Program., Ser. B 117(1–2), 5–19 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Drusvyatskiy, D., Ioffe, A.D., Lewis, A.S.: The dimension of semialgebraic subdifferential graphs. Nonlinear Anal. 75(3), 1231–1245 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Bochnak, J., Coste, M., Roy, M.-F.: Géométrie Algébrique Réelle. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 12 [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1987) zbMATHGoogle Scholar
  17. 17.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 317. Springer, Berlin (1998) CrossRefzbMATHGoogle Scholar
  18. 18.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I Basic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006) Google Scholar
  19. 19.
    Rockafellar, R.T.: Favorable classes of Lipschitz-continuous functions in subgradient optimization. In: Nurminski, E.A. (ed.) Progress in Nondifferentiable Optimization. IIASA Collaborative Proc. Ser. CP-82, vol. S8, pp. 125–143. Internat. Inst. Appl. Systems Anal. (IIASA), Laxenburg (1982) Google Scholar
  20. 20.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Classics in Applied Mathematics, vol. 5. SIAM, Philadelphia (1990). First edition published in Wiley, New York (1983) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Chair of Applied MathematicsUniversity of BayreuthBayreuthGermany
  2. 2.School of Mathematical Sciences, Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Collaborative Research NetworkUniversity of BallaratBallaratAustralia

Personalised recommendations