On global subdifferentials with applications in nonsmooth optimization

Abstract

The notions of global subdifferentials associated with the global directional derivatives are introduced in the following paper. Most common used properties, a set of calculus rules along with a mean value theorem are presented as well. In addition, a diversity of comparisons with well-known subdifferentials such as Fréchet, Dini, Clarke, Michel–Penot, and Mordukhovich subdifferential and convexificator notion are provided. Furthermore, the lower global subdifferential is in fact proved to be an abstract subdifferential. Therefore, the lower global subdifferential satisfies standard properties for subdifferential operators. Finally, two applications in nonconvex nonsmooth optimization are given: necessary and sufficient optimality conditions for a point to be local minima with and without constraints, and a revisited characterization for nonsmooth quasiconvex functions.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Ansari, Q.H., Lalitha, C.S., Mehta, M.: Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization. CRC Press, Boca Raton (2014)

    Google Scholar 

  2. 2.

    Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2003)

    Google Scholar 

  3. 3.

    Auslender, A., Teboulle, M.: Interior gradient and epsilon-subgradient descent methods for constrained convex optimization. Math. Oper. Res. 29, 1–26 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Aussel, D.: Subdifferentials properties of quasiconvex and pseudoconvex functions: unified approach. J. Optim. Theory Appl. 97, 29–45 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Aussel, D., Corvellec, J.N., Lassonde, M.: Mean-value property and subdifferential criteria for lower semicontinuous functions. Trans. Am. Math. Soc. 347, 4147–4161 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Arrow, K.J., Enthoven, A.C.: Quasiconcave programming. Econometrica 29, 779–800 (1961)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Cambini, A., Martein, L.: Generalized Convexity and Optimization. Springer, Berlin, Heidelberg (2009)

    Google Scholar 

  8. 8.

    Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, New York (1990)

    Google Scholar 

  9. 9.

    Crouzeix J.P.: Some properties of Dini-derivatives of quasiconvex functions. In: Giannessi, F., et al. (eds.): New Trends in Mathematical Programming. Applied Optimization, vol. 13, pp. 41–57. Springer, Boston (1998)

  10. 10.

    Da Cruz Neto, J.X., Da Silva, G., Ferreira, O., Lopez, J.: A subgradient method for multiobjective optimization. Comput. Optim. Appl. 54, 461–472 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Giorgi, G., Guerraggio, G., Thierfelder, T.: Mathematics of Optimization: Smooth and Nonsmooth Case. Elsevier, Amsterdam (2004)

    Google Scholar 

  12. 12.

    Giorgi, G., Komlosi, S.: Dini derivatives in optimization—part I. Riv. Mat. Sci. Econom. Soc. 15(1), 3–30 (1993)

    MATH  Google Scholar 

  13. 13.

    Giorgi, G., Komlosi, S.: Dini derivatives in optimization—part II. Riv. Mat. Sci. Econom. Soc. 15(2), 3–24 (1993)

    MATH  Google Scholar 

  14. 14.

    Hadjisavvas, N., Komlosi, S., Schaible, S.: Handbook of Generalized Convexity and Generalized Monotonicity. Springer, Boston (2005)

    Google Scholar 

  15. 15.

    Hiriart-Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993)

    Google Scholar 

  16. 16.

    Ioffe, A.D.: Approximate subdifferentials and applications I: the finite dimensional theory. Trans. Am. Soc. 281, 389–416 (1984)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Jeyakumar, V., Luc, D.T.: Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory Appl. 101, 599–621 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Kabgani, A., Soleimani-damaneh, M.: Relationship between convexificators and Greenberg–Pierskalla subdifferentials for quasiconvex functions. Numer. Funct. Anal. Optim. 38, 1548–1563 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Kabgani, A., Soleimani-damaneh, M., Zamani, M.: Optimality conditions in optimization problems with convex feasible set using convexificators. Math. Methods Oper. Res. 86, 103–121 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Kim, S., Um, B.: An improved subgradient method for constrained nondifferentiable optimization. Oper. Res. Lett. 14, 61–64 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Lara, F.: Optimality conditions for nonconvex nonsmooth optimization via global derivatives. J. Optim. Theory Appl. 185, 134–150 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Linh, N.T.H., Penot, J.P.: Optimality conditions for quasiconvex programs. SIAM J. Optim. 17, 500–510 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Linh, N.T.H., Penot, J.P.: Generalized convex functions and generalized differentials. Optimization 62, 934–959 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Luc, D.T., Théra, M.: Derivatives with support and applications. Math. Oper. Res. 19, 659–675 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Michel, P., Penot, J.P.: Calcul sous-différentiel pour des fonctions lipschitziennes et non lipschitziennes. CR Acad. Sci. Paris 298, 269–272 (1984)

    MATH  Google Scholar 

  26. 26.

    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  27. 27.

    Penot J.P.: Are generalized derivatives useful for generalized convex functions?. In: Crouzeix, J.P., et al. (eds.): Generalized Convexity, Generalized Monotonicity, pp. 3–60. Kluwer, Amsterdam (1998)

  28. 28.

    Penot, J.P.: What is quasiconvex analysis? Optimization 47, 35–110 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Google Scholar 

  30. 30.

    Rockafellar, R.T.: Generalized directional derivatives and subgradients of nonconvex functions. Can. J. Math. 32, 257–280 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Soleimani-Damaneh, M.: Characterization of nonsmooth quasiconvex and pseudoconvex functions. J. Math. Anal. Appl. 330, 1387–1392 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Wu, Z., Ye, J.J.: Equivalence among various derivatives and subdifferentials of the distance function. J. Math. Anal. Appl. 282(2), 629–647 (2003)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

This research was partially supported by Conicyt–Chile under project Fondecyt Iniciación 11180320 (Lara) and by IPM under the grant number 98900032 (Kabgani).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Felipe Lara.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lara, F., Kabgani, A. On global subdifferentials with applications in nonsmooth optimization. J Glob Optim (2021). https://doi.org/10.1007/s10898-020-00981-1

Download citation

Keywords

  • Nonsmooth analysis
  • Global derivatives
  • Global subdifferentials
  • Nonconvex optimization
  • Local minima

Mathematics Subject Classification

  • 90C30
  • 90C26