Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In the research of optimization problems, optimality conditions play an important role. By using some derivatives, various types of necessary and/or sufficient optimality conditions have been introduced by many researchers. Especially, in convex programming, necessary and sufficient optimality conditions in terms of the subdifferential have been studied extensively. Recently, necessary and sufficient optimality conditions for quasiconvex programming have been investigated by the authors. However, there are not so many results concerned with Karush–Kuhn–Tucker type optimality conditions for non-differentiable quasiconvex programming. In this paper, we study a Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. We show some closedness properties for Greenberg–Pierskalla subdifferential. Under the Slater constraint qualification, we show a necessary and sufficient optimality condition for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. Additionally, we introduce a necessary and sufficient constraint qualification of the optimality condition. As a corollary, we show a necessary and sufficient optimality condition for convex programming in terms of the subdifferential.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Mathematical Concepts and Methods in Science and Engineering. Plenum Press, New York (1988)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  9. Suzuki, S.: Duality theorems for quasiconvex programming with a reverse quasiconvex constraint. Taiwanese J. Math. 21, 489–503 (2017)

    Article  MathSciNet  Google Scholar 

  10. Suzuki, S.: Optimality conditions and constraint qualifications for quasiconvex programming. J. Optim. Theory Appl. 183, 963–976 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. 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)

    Article  MathSciNet  Google Scholar 

  14. Suzuki, S., Kuroiwa, D.: Characterizations of the solution set for non-essentially quasiconvex programming. Optim. Lett. 11, 1699–1712 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Al-Homidan, S., Hadjisavvas, N., Shaalan, L.: Transformation of quasiconvex functions to eliminate local minima. J. Optim. Theory Appl. 177, 93–105 (2018)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Martínez-Legaz, J.E., Sach, P.H.: A new subdifferential in quasiconvex analysis. J. Convex Anal. 6, 1–11 (1999)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Cambini, A., Martein, L.: Generalized Convexity and Optimization Theory and Applications. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2009)

    MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  Google Scholar 

  29. Jeyakumar, V.: Constraint qualifications characterizing Lagrangian duality in convex optimization. J. Optim. Theory Appl. 136, 31–41 (2008)

    Article  MathSciNet  Google Scholar 

  30. 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)

  31. Mangasarian, O.L.: Set containment characterization. J. Glob. Optim. 24, 473–480 (2002)

    Article  MathSciNet  Google Scholar 

  32. Suzuki, S., Kuroiwa, D.: On set containment characterization and constraint qualification for quasiconvex programming. J. Optim. Theory Appl. 149, 554–563 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Suzuki, S., Kuroiwa, D.: Generators and constraint qualifications for quasiconvex inequality systems. J. Nonlinear Convex Anal. 18, 2101–2121 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Suzuki, S., Kuroiwa, D.: Duality theorems for separable convex programming without qualifications. J. Optim. Theory Appl. 172, 669–683 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Shimane University, Matsue, Shimane, Japan

    Satoshi Suzuki

Authors
  1. Satoshi Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Satoshi Suzuki.

Additional information

Publisher's Note

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

This work was partially supported by JSPS KAKENHI Grant Number 19K03620.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suzuki, S. Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. J Glob Optim 79, 191–202 (2021). https://doi.org/10.1007/s10898-020-00926-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-020-00926-8

Keywords

Search

Navigation