Characterization of Radially Lower Semicontinuous Pseudoconvex Functions

  • Vsevolod I. IvanovEmail author


In this paper, we prove that a radially lower semicontinuous function of one variable, defined on some interval, is pseudoconvex, if and only if its domain of definition can be split into three parts such that the function is strictly monotone decreasing without stationary points over the first subinterval, it is constant over the second one, and it is strictly monotone increasing without stationary points over the third subinterval. Each one or two of these parts may be empty or degenerate into a single point. The proof of this property is easy, when the function is differentiable. We consider functions, which are pseudoconvex with respect to the lower Dini directional derivative. This result follows from some known claims, but our theorem is a shot, directed to the target. We apply this characterization to obtain a complete characterization of strictly pseudoconvex functions. We also derive the respective results, when the function is radially lower semicontinuous in a real linear space. Several applications of the characterization are provided. A result due to Diewert, Avriel and Zang is extended to radially continuous functions.


Pseudoconvex function Nonsmooth analysis Nonsmooth optimization Lower Dini directional derivative 

Mathematics Subject Classification

26B25 49J52 90C26 



The author would like to express his gratitude to the anonymous referees for their helpful comments on the manuscript. This research is partially supported by the TU-Varna Grant No. 19/2019.


  1. 1.
    Avriel, M., Diewert, W., Schaible, S. Zang, I.: Generalized Concavity. Classics in Applied Mathematics, No. 63, SIAM, Philadelphia (2010), Originally published: Plenum Press, New York (1988)Google Scholar
  2. 2.
    Bazaraa, M.S., Sheraly, H.D., Shetty, C.M.: Nonlinear Programming—Theory and Algorithms. Wiley, New York (2006)CrossRefGoogle Scholar
  3. 3.
    Cambini, A., Martein, L.: Generalized Convexity and Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 616. Springer, Berlin (2009)zbMATHGoogle Scholar
  4. 4.
    Giorgi, G., Guerraggio, A., Thierfelder, J.: Mathematics of Optimization. Elsevier Science, New York (2004)zbMATHGoogle Scholar
  5. 5.
    Mangasarian, O.L.: Nonlinear Programming. Classics in Applied Mathematics, vol. 10. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  6. 6.
    Martos, B.: Nonlinear Programming—Theory and Methods. Akademiai Kiado, Budapest (1975)zbMATHGoogle Scholar
  7. 7.
    Mangasarian, O.L.: Pseudo-convex functions. SIAM J. Control 3, 281–290 (1965)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Giorgi, G.: Optimality conditions under generalized convexity revisited. Ann. Univ. Buchar. Math. Ser. 4, 479–490 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Barani, A.: Convexity of the solution set of a pseudoconvex inequality on Riemannian manifolds. Numer. Funct. Anal. Optim. 39, 588–599 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ivanov, V.I.: First-order characterizations of pseudoconvex functions. Serdica Math. J. 27, 203–218 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ivanov, V.I.: Optimality conditions and characterizations of the solution sets in generalized convex problems and variational inequalities. J. Optim. Theory Appl. 158, 65–84 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ivanov, V.I.: Characterizations of pseudoconvex functions and semistrictly quasiconvex ones. J. Glob. Optim. 57, 677–693 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ivanov, V.I.: Characterizations of solution sets of differentiable quasiconvex programming problems. J. Optim. Theory Appl. 181, 144–162 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jaddar, A.: On optimality conditions for pseudoconvex programming in terms of Dini subdifferentials. Int. J. Math. Anal. 7, 891–898 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Son, T.Q., Kim, D.S.: A new approach to characterize solution set of a pseudoconvex programming problem. J. Comput. Appl. Math. 261, 333–340 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Suzuki, S., Kuroiwa, D.: Characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. J. Glob. Optim. 62, 431–441 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Levi, E.E.: Studi sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse. Ann. Mat. Pura Appl. 17, 61–68 (1910)CrossRefGoogle Scholar
  18. 18.
    Avriel, M., Schaible, S.: Second-order characterizations of pseudoconvex functions. Math. Program. 14, 170–185 (1978)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Aussel, D.: Subdifferential properties of quasiconvex and pseudoconvex functions: unified approach. J. Optim. Theory Appl. 97, 29–45 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Crouzeix, J.P., Ferland, J.: Criteria for quasiconvexity and pseudoconvexity: relations and comparisons. Math. Program. 23, 193–202 (1982)CrossRefGoogle Scholar
  21. 21.
    Daniilidis, A., Hadjisavvas, N.: On the subdifferentials of quasiconvex and pseudoconvex functions and cyclic monotonicity. J. Math. Anal. Appl. 237, 30–42 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Diewert, W.E.: Alternative characterizations of six kinds of quasiconvexity in the nondifferentiable case with applications to nonlinear programming. In: Schaible, S., Ziemba, W.T. (eds.) Generalized Concavity in Optimization and Economics, pp. 51–93. Academic Press, New York (1981)zbMATHGoogle Scholar
  23. 23.
    Diewert, W.E., Avriel, M., Zang, I.: Nine kinds of quasiconcavity and concavity. J. Econ. Theory 25, 397–420 (1981)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ginchev, I., Ivanov, V.I.: Second-order characterizations of convex and pseudoconvex functions. J. Appl. Anal. 9, 261–273 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Giorgi, G., Thierfelder, J.: Constrained quadratic forms and generalized convexity of \({\rm C}^2\)-functions revisited. In: Giorgi, G., Rossi, F. (eds.) Generalized Convexity and Optimization for Economic and Financial Decisions, pp. 179–219. Pitagora Editrice, Bologna (1999)zbMATHGoogle Scholar
  26. 26.
    Hassouni, A., Jaddar, A.: On pseudoconvex functions and applications to global optimization. ESAIM Proc. 20, 128–148 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Khanh, P.D., Phat, V.T.: On second-order conditions for quasiconvexity and pseudoconvexity of \({\rm C}^{1,1}\)-smooth functions. arXiv:1810.12783 (2018)
  28. 28.
    Komlosi, S.: Some properties of nondifferentiable pseudoconvex functions. Math. Program. 26, 232–237 (1983)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Komlosi, S.: On pseudoconvex functions. Acta Math. Sci. (Szeged) 57, 569–586 (1993)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Komlosi, S.: Generalized monotonicity and generalized convexity. J. Optim. Theory Appl. 84, 361–376 (1995)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Mereau, P., Paquet, J.-G.: Second order conditions for pseudo-convex functions. SIAM J. Appl. Math. 27, 131–136 (1974)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Miflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Penot, J.-P., Quang, P.H.: Generalized convexity of functions and generalized monotonicity of set-valued maps. J. Optim. Theory Appl. 92, 343–356 (1997)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Soleimani-damaneh, M.: Characterizations of nonsmooth quasiconvex and pseudoconvex functions. J. Math. Anal. Appl. 330, 1387–1392 (2007)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Soleimani-damaneh, M.: On generalized convexity in Asplund spaces. Nonlinear Anal. 70, 3072–3075 (2009)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Thompson, W., Parke, D.: Some properties of generalized concave functions. Oper. Res. 21, 305–313 (1973)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  38. 38.
    Pshenichnyi, B.N.: Necessary Conditions for Extremum. Marcel Dekker, New York (1983)zbMATHGoogle Scholar
  39. 39.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of VarnaVarnaBulgaria

Personalised recommendations