Skip to main content
SpringerLink
Log in

Necessary and Sufficient Conditions for Strong Fenchel–Lagrange Duality via a Coupling Conjugation Scheme

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Given a general primal problem and its Fenchel–Lagrange dual one, which is obtained by using a conjugation scheme based on coupling functions and the perturbational approach, the aim in this work is to establish conditions under which strong duality can be guaranteed. To this purpose, even convexity and properness are a compulsory requirement over the involved functions in the primal problem. Furthermore, two closedness-type regularity conditions and a characterization for strong duality are derived.

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. Rockafellar, R.T.: Conjugate duality and optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA. (1974). Lectures given at the Johns Hopkins University, Baltimore, MD, June, 1973

  2. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Co., Amsterdam (1976)

    MATH  Google Scholar 

  3. Fenchel, W.: A remark on convex sets and polarity (Tome Supplémentaire), 82–89 (1952)

  4. Rodríguez, M.M.L., Vicente-Pérez, J.: On evenly convex functions. J. Convex Anal. 18(3), 721–736 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Fajardo, M.D.: Regularity conditions for strong duality in evenly convex optimization problems. An application to Fenchel duality. J. Convex Anal. 22(3), 711–731 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Martínez-Legaz, J.E., Vicente-Pérez, J.: The e-support function of an e-convex set and conjugacy for e-convex functions. J. Math. Anal. Appl. 376(2), 602–612 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. Journal de Mathématiques Pures et Appliquées. Neuvième Série 49, 109–154 (1970)

    MathSciNet  MATH  Google Scholar 

  8. Fajardo, M.D., Vicente-Pérez, J., Rodríguez, M.M.L.: Infimal convolution, \(c\)-subdifferentiability, and Fenchel duality in evenly convex optimization. TOP 20(2), 375–396 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fajardo, M.D., Rodríguez, M.M.L., Vidal, J.: Lagrange duality for evenly convex optimization problems. J. Optim. Theory Appl. 168(1), 109–128 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wanka, G., Boţ, R.I.: On the relations between different dual problems in convex mathematical programming. In: Operations Research Proceedings, 2001 (Duisburg), pp. 255–262. Springer, Berlin (2002)

  11. Boţ, R.I., Grad, S.M., Wanka, G.: New regularity conditions for strong and total Fenchel–Lagrange duality in infinite dimensional spaces. Nonlinear Anal. Theory Methods Appl. 69(1), 323–336 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Boţ, R.I., Grad, S.M., Wanka, G.: New regularity conditions for Lagrange and Fenchel–Lagrange duality in infinite dimensional spaces. Math. Inequal. Appl. 12(1), 171–189 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Boţ, R.I., Kassay, G., Wanka, G.: Strong duality for generalized convex optimization problems. J. Optim. Theory Appl. 127(1), 45–70 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Boţ, R.I., Grad, S.M., Wanka, G.: Fenchel–Lagrange duality versus geometric duality in convex optimization. J. Optim. Theory Appl. 129(1), 33–54 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Boţ, R.I., Grad, S.M., Wanka, G.: A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces. Math. Nachr. 281(8), 1088–1107 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Boţ, R.I., Grad, S.M., Wanka, G.: Generalized Moreau–Rockafellar results for composed convex functions. Optimization 58(7), 917–933 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Boţ, R.I., Grad, S.M., Wanka, G.: Duality in Vector Optimization. Vector Optimization. Springer, Berlin (2009)

    MATH  Google Scholar 

  18. Aboussoror, A., Adly, S.: A Fenchel–Lagrange duality approach for a bilevel programming problem with extremal-value function. J. Optim. Theory Appl. 149(2), 254–268 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhou, Y., Li, G.: The Toland–Fenchel–Lagrange duality of DC programs for composite convex functions. Numer. Algebra Control Optim. 4(1), 9–23 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Sun, X.K.: Regularity conditions characterizing Fenchel–Lagrange duality and Farkas-type results in DC infinite programming. J. Math. Anal. Appl. 414(2), 590–611 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Aboussoror, A., Adly, S., Saissi, F.E.: An extended Fenchel–Lagrange duality approach and optimality conditions for strong bilevel programming problems. SIAM J. Optim. 27(2), 1230–1255 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fajardo, M.D., Vidal, J.: A comparison of alternative c-conjugate dual problems in infinite convex optimization. Optimization 66(5), 705–722 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Daniilidis, A., Martínez-Legaz, J.E.: Characterizations of evenly convex sets and evenly quasiconvex functions. J. Math. Anal. Appl. 273(1), 58–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Goberna, M.A., Jornet, V., Rodríguez, M.M.L.: On linear systems containing strict inequalities. Linear Algebra Appl. 360, 151–171 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.): Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications, vol. 76. Springer, New York (2005)

    MATH  Google Scholar 

  26. Fajardo, M.D., Vidal, J.: Stable strong Fenchel and Lagrange duality for evenly convex optimization problems. Optimization 65(9), 1675–1691 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Grad, S.M.: Closedness type regularity conditions in convex optimization and beyond. Front. Appl. Math. Stat. 2, 14 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

This research was partially supported by MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P and by Consellería de Educación de la Generalitat Valenciana, Spain, Pre-doc Program Vali+d, DOCV 6791/07.06.2012, Grant ACIF-2013-156. The authors wish to thank anonymous referee for her/his valuable comments and suggestions that have significantly improved the quality of the paper.

Author information

Authors and Affiliations

  1. University of Alicante, Alicante, Spain

    M. D. Fajardo

  2. Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany

    J. Vidal

Authors
  1. M. D. Fajardo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Vidal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. D. Fajardo.

Additional information

Vaithilingam Jeyakumar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fajardo, M.D., Vidal, J. Necessary and Sufficient Conditions for Strong Fenchel–Lagrange Duality via a Coupling Conjugation Scheme. J Optim Theory Appl 176, 57–73 (2018). https://doi.org/10.1007/s10957-017-1209-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1209-x

Keywords

Mathematics Subject Classification

Search

Navigation