Skip to main content
SpringerLink
Log in

Characterizations of the solution set for non-essentially quasiconvex programming

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

Characterizations of the solution set in terms of subdifferentials play an important role in research of mathematical programming. Previous characterizations are based on necessary and sufficient optimality conditions and invariance properties of subdifferentials. Recently, characterizations of the solution set for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential are studied by the authors. Unfortunately, there are some examples such that these characterizations do not hold for non-essentially quasiconvex programming. As far as we know, characterizations of the solution set for non-essentially quasiconvex programming have not been studied yet. In this paper, we study characterizations of the solution set in terms of subdifferentials for non-essentially quasiconvex programming. For this purpose, we use Martínez–Legaz subdifferential which is introduced by Martínez–Legaz as a special case of c-subdifferential by Moreau. We derive necessary and sufficient optimality conditions for quasiconvex programming by means of Martínez–Legaz subdifferential, and, as a consequence, investigate characterizations of the solution set in terms of Martínez–Legaz subdifferential. In addition, we compare our results with previous ones. We show an invariance property of Greenberg–Pierskalla subdifferential as a consequence of an invariance property of Martínez–Legaz subdifferential. We give characterizations of the solution set for essentially quasiconvex programming in terms of Martínez–Legaz subdifferential.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity. Math. Concepts Methods Sci. Engrg. Plenum Press, New York (1988)

    MATH  Google Scholar 

  2. Burke, J.V., Ferris, M.C.: Characterization of solution sets of convex programs. Oper. Res. Lett. 10, 57–60 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Crouzeix, J.P., Ferland, J.A.: Criteria for quasiconvexity and pseudoconvexity: relationships and comparisons. Math. Program. 23, 193–205 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  4. Daniilidis, A., Hadjisavvas, N., Martínez-Legaz, J.E.: An appropriate subdifferential for quasiconvex functions. SIAM J. Optim. 12, 407–420 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fang, D.H., Luo, X.F., Wang, X.Y.: Strong and total lagrange dualities for quasiconvex programming. J. Appl. Math. doi:10.1155/2014/453912

  6. Greenberg, H.J., Pierskalla, W.P.: Quasi-conjugate functions and surrogate duality. Cah. Cent. Étud. Rech. Opér. 15, 437–448 (1973)

    MATH  MathSciNet  Google Scholar 

  7. Gutiérrez Díez, J.M.: Infragradientes \(y\) Direcciones de Decrecimiento. Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid. 78, 523–532 (1984)

  8. Hu, Y., Yang, X., Sim, C.K.: Inexact subgradient methods for quasi-convex optimization problems. Eur. J. Oper. Res. 240, 315–327 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ivanov, V.I.: First order characterizations of pseudoconvex functions. Serdica Math. J. 27, 203–218 (2001)

    MATH  MathSciNet  Google Scholar 

  10. Ivanov, V.I.: Characterizations of the solution sets of generalized convex minimization problems. Serdica Math. J. 29, 1–10 (2003)

    MATH  MathSciNet  Google Scholar 

  11. Ivanov, V.I.: Characterizations of pseudoconvex functions and semistrictly quasiconvex ones. J. Global Optim. 57, 677–693 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jeyakumar, V., Lee, G.M., Dinh, N.: Lagrange multiplier conditions characterizing the optimal solution sets of cone-constrained convex programs. J. Optim. Theory Appl. 123, 83–103 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21–26 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Martínez-Legaz, J.E.: A generalized concept of conjugation. Lecture Notes Pure Appl. Math. 86, 45–59 (1983)

    MATH  MathSciNet  Google Scholar 

  17. Martínez-Legaz, J.E.: A new approach to symmetric quasiconvex conjugacy. Lecture Notes Econ. Math. Syst. 226, 42–48 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  18. Martínez-Legaz, J.E.: Quasiconvex duality theory by generalized conjugation methods. Optimization 19, 603–652 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  19. Martínez-Legaz, J.E., Sach, P.H.: A New Subdifferential in Quasiconvex Analysis. J. Convex Anal. 6, 1–11 (1999)

    MATH  MathSciNet  Google Scholar 

  20. Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Penot, J.P.: Characterization of solution sets of quasiconvex programs. J. Optim. Theory Appl. 117, 627–636 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Penot, J.P., Volle, M.: On quasi-convex duality. Math. Oper. Res. 15, 597–625 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  24. Plastria, F.: Lower subdifferentiable functions and their minimization by cutting planes. J. Optim. Theory Appl. 46, 37–53 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  25. 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)

    Article  MATH  MathSciNet  Google Scholar 

  26. Suzuki, S., Kuroiwa, D.: Optimality conditions and the basic constraint qualification for quasiconvex programming. Nonlinear Anal. 74, 1279–1285 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  27. Suzuki, S., Kuroiwa, D.: Subdifferential calculus for a quasiconvex function with generator. J. Math. Anal. Appl. 384, 677–682 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  28. Suzuki, S., Kuroiwa, D.: Necessary and sufficient conditions for some constraint qualifications in quasiconvex programming. Nonlinear Anal. 75, 2851–2858 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. Suzuki, S., Kuroiwa, D.: Necessary and sufficient constraint qualification for surrogate duality. J. Optim. Theory Appl. 152, 366–367 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Suzuki, S., Kuroiwa, D.: Some constraint qualifications for quasiconvex vector-valued systems. J. Global Optim. 55, 539–548 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. 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)

    Article  MATH  MathSciNet  Google Scholar 

  32. Suzuki, S., Kuroiwa, D.: Generators and constraint qualifications for quasiconvex inequality systems. J. Nonlinear Convex Anal. (2016) (to appear)

  33. 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)

    Article  MATH  MathSciNet  Google Scholar 

  34. Yang, X.M.: On characterizing the solution sets of pseudoinvex extremum problems. J. Optim. Theory Appl. 140, 537–542 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Zhao, K.Q., Yang, X.M.: Characterizations of the solution set for a class of nonsmooth optimization problems. Optim. Lett. 7, 685–694 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are grateful to anonymous referees for many comments and suggestions improved the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shimane University, 1060 Nishikawatsu, Matsue, Shimane, Japan

    Satoshi Suzuki & Daishi Kuroiwa

Authors
  1. Satoshi Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daishi Kuroiwa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Satoshi Suzuki.

Additional information

This work was partially supported by JSPS KAKENHI Grant Numbers 15K17588, 16K05274.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suzuki, S., Kuroiwa, D. Characterizations of the solution set for non-essentially quasiconvex programming. Optim Lett 11, 1699–1712 (2017). https://doi.org/10.1007/s11590-016-1084-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-016-1084-7

Keywords

Search

Navigation