Optimality Conditions and Constraint Qualifications for Quasiconvex Programming

  • Satoshi SuzukiEmail author
Regular Paper


In mathematical programming, various kinds of optimality conditions have been introduced. In the research of optimality conditions, some types of subdifferentials play an important role. Recently, by using Greenberg–Pierskalla subdifferential and Martínez-Legaz subdifferential, necessary and sufficient optimality conditions for quasiconvex programming have been introduced. On the other hand, constraint qualifications are essential elements for duality theory in mathematical programming. Over the last decade, necessary and sufficient constraint qualifications for duality theorems have been investigated extensively. Recently, by using the notion of generator, necessary and sufficient constraint qualifications for Lagrange-type duality theorems have been investigated. However, constraint qualifications for optimality conditions in terms of Greenberg–Pierskalla subdifferential and Martínez-Legaz subdifferential have not been investigated yet. In this paper, we study optimality conditions and constraint qualifications for quasiconvex programming. We introduce necessary and sufficient optimality conditions in terms of Greenberg–Pierskalla subdifferential, Martínez-Legaz subdifferential and generators. We investigate necessary and/or sufficient constraint qualifications for these optimality conditions. Additionally, we show some equivalence relations between duality results for convex and quasiconvex programming.


Quasiconvex programming Optimality condition Constraint qualification Generator of a quasiconvex function 

Mathematics Subject Classification

90C26 90C46 49J52 



The author is grateful to anonymous referees for many comments and suggestions which improved the quality of the paper.


  1. 1.
    Boţ, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin, (2010)Google Scholar
  2. 2.
    Burke, J.V., Ferris, M.C.: Characterization of solution sets of convex programs. Oper. Res. Lett. 10, 57–60 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Li, C., Ng, K.F., Pong, T.K.: Constraint qualifications for convex inequality systems with applications in constrained optimization. SIAM J. Optim. 19, 163–187 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21–26 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  6. 6.
    Wu, Z.L., Wu, S.Y.: Characterizations of the solution sets of convex programs and variational inequality problems. J. Optim. Theory Appl. 130, 339–358 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ivanov, V.I.: Characterizations of the solution sets of generalized convex minimization problems. Serdica Math. J. 29, 1–10 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ivanov, V.I.: Characterizations of pseudoconvex functions and semistrictly quasiconvex ones. J. Global Optim. 57, 677–693 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Son, T.Q., Kim, D.S.: A new approach to characterize the solution set of a pseudoconvex programming problem. J. Comput. Appl. Math. 261, 333–340 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yang, X.M.: On characterizing the solution sets of pseudoinvex extremum problems. J. Optim. Theory Appl. 140, 537–542 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Zhao, K.Q., Yang, X.M.: Characterizations of the solution set for a class of nonsmooth optimization problems. Optim. Lett. 7, 685–694 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized concavity. Math. Concepts Methods Sci. Engrg. Plenum Press, New York (1988)Google Scholar
  14. 14.
    Ivanov, V.I.: Characterizations of solution sets of differentiable quasiconvex programming problems. J. Optim. Theory Appl.
  15. 15.
    Linh, N.T.H., Penot, J.P.: Optimality conditions for quasiconvex programs. SIAM J. Optim. 17, 500–510 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Penot, J.P.: Characterization of solution sets of quasiconvex programs. J. Optim. Theory Appl. 117, 627–636 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Suzuki, S., Kuroiwa, D.: Optimality conditions and the basic constraint qualification for quasiconvex programming. Nonlinear Anal. 74, 1279–1285 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Suzuki, S., Kuroiwa, D.: Subdifferential calculus for a quasiconvex function with generator. J. Math. Anal. Appl. 384, 677–682 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Suzuki, S., Kuroiwa, D.: Some constraint qualifications for quasiconvex vector-valued systems. J. Global Optim. 55, 539–548 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Suzuki, S., Kuroiwa, D.: Characterizations of the solution set for quasiconvex programming in terms of Greenberg-Pierskalla subdifferential. J. Global Optim. 62, 431–441 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Suzuki, S., Kuroiwa, D.: Characterizations of the solution set for non-essentially quasiconvex programming. Optim. Lett. 11, 1699–1712 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Suzuki, S.: Duality theorems for quasiconvex programming with a reverse quasiconvex constraint. Taiwanese J. Math. 21, 489–503 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Goberna, M.A., Jeyakumar, V., López, M.A.: Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities. Nonlinear Anal. 68, 1184–1194 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jeyakumar, V.: Constraint qualifications characterizing Lagrangian duality in convex optimization. J. Optim. Theory Appl. 136, 31–41 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jeyakumar, V., Dinh, N., Lee, G. M.: A new closed cone constraint qualification for convex optimization. Research Report AMR 04/8, Department of Applied Mathematics, University of New South Wales, (2004)Google Scholar
  26. 26.
    Suzuki, S., Kuroiwa, D.: On set containment characterization and constraint qualification for quasiconvex programming. J. Optim. Theory Appl. 149, 554–563 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Suzuki, S., Kuroiwa, D.: Necessary and sufficient conditions for some constraint qualifications in quasiconvex programming. Nonlinear Anal. 75, 2851–2858 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Suzuki, S., Kuroiwa, D.: Necessary and sufficient constraint qualification for surrogate duality. J. Optim. Theory Appl. 152, 366–367 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Suzuki, S., Kuroiwa, D.: Generators and constraint qualifications for quasiconvex inequality systems. J. Nonlinear Convex Anal. 18, 2101–2121 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Suzuki, S., Kuroiwa, D.: Duality Theorems for Separable Convex Programming without Qualifications. J. Optim. Theory Appl. 172, 669–683 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Martínez-Legaz, J.E.: Quasiconvex duality theory by generalized conjugation methods. Optimization. 19, 603–652 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Penot, J.P., Volle, M.: On quasi-convex duality. Math. Oper. Res. 15, 597–625 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Crouzeix, J.P., Ferland, J.A.: Criteria for quasiconvexity and pseudoconvexity: relationships and comparisons. Math. Programming. 23, 193–205 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ivanov, V.I.: First order characterizations of pseudoconvex functions. Serdica Math. J. 27, 203–218 (2001)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Al-Homidan, S., Hadjisavvas, N., Shaalan, L.: Transformation of quasiconvex functions to eliminate local minima. J. Optim. Theory Appl. 177, 93–105 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Greenberg, H.J., Pierskalla, W.P.: Quasi-conjugate functions and surrogate duality. Cah. Cent. Étud. Rech. Opér. 15, 437–448 (1973)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Daniilidis, A., Hadjisavvas, N., Martínez-Legaz, J.E.: An appropriate subdifferential for quasiconvex functions. SIAM J. Optim. 12, 407–420 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Gutiérrez Díez, J.M.: Infragradientes \(y\) Direcciones de Decrecimiento. Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid. 78, 523–532 (1984)MathSciNetGoogle Scholar
  39. 39.
    Hu, Y., Yang, X., Sim, C.K.: Inexact subgradient methods for quasi-convex optimization problems. European J. Oper. Res. 240, 315–327 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Martínez-Legaz, J.E.: A generalized concept of conjugation. Lecture Notes in Pure and Appl. Math. 86, 45–59 (1983)Google Scholar
  41. 41.
    Martínez-Legaz, J.E.: A new approach to symmetric quasiconvex conjugacy. Lecture Notes in Econom. and Math. Systems. 226, 42–48 (1984)Google Scholar
  42. 42.
    Martínez-Legaz, J.E., Sach, P.H.: A new subdifferential in quasiconvex analysis. J. Convex Anal. 6, 1–11 (1999)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Penot, J.P.: What is quasiconvex analysis? Optimization. 47, 35–110 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Plastria, F.: Lower subdifferentiable functions and their minimization by cutting planes. J. Optim. Theory Appl. 46, 37–53 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.Department of Mathematical SciencesShimane UniversityShimaneJapan

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