Skip to main content
SpringerLink
Log in

Dual method of centers for solving generalized fractional programs

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

Abstract

In this paper we analyze the method of centers for generalized fractional programs, with further insights. The introduced method is based on a different parametric auxiliary problem than Dinkelbach’s type. With the help of this auxiliary parametric problem, we present a new dual for convex generalized fractional programs. We then propose an algorithm to solve this problem, and subsequently the original primal program. The proposed algorithm generates a sequence of dual values that converges from below to the optimal value. The method also generates a bounded sequence of dual solutions whose every accumulation point is a solution of the dual problem. The rate of convergence is shown to be at least linear. In the penultimate section, we specialize the results obtained for the linear case. The computational results show that the different variants of our algorithms, primal as well as dual, are competitive.

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. Addou, A., Roubi, A.: Proximal-type methods with generalized Bregman functions and applications to generalized fractional programming. Optimization 59(7), 1085–1105 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barros, A.I., Frenk, J.B.G., Schaible, S., Zhang, S.: Generalized fractional programming and cutting plane algorithms. J. Optim. Theory Appl. 87(1), 103–120 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barros, A.I., Frenk, J.B.G., Schaible, S., Zhang, S.: A new algorithm for generalized fractional programs. Math. Program. 72, 147–175 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Barros, A.I., Frenk, J.B.G., Schaible, S., Zhang, S.: Using duality to solve generalized fractional programming problems. J. Global Optim. 8, 139–170 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bector, C.R., Chandra, S., Bector, M.K.: Generalized fractional programming duality: a parametric approach. J. Optim. Theory Appl. 60(2), 243–260 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benadada, Y., Ferland, J.A.: Partial linearization for generalized fractional programming. Zeitschrift für Oper. Res. 32, 101–106 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Bernard, J.C., Ferland, J.A.: Convergence of interval-type algorithms for generalized fractional programming. Math. Program. 43, 349–363 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Crouzeix, J.P., Ferland, J.A.: Algorithms for generalized fractional programming. Math. Program. 52, 191–207 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Crouzeix, J.P., Ferland, J.A., Schaible, S.: Duality in generalized linear fractional programming. Math. Program. 27, 342–354 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Crouzeix, J.P., Ferland, J.A., Schaible, S.: An algorithm for generalized fractional programs. J. Optim. Theory Appl. 47, 35–49 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Crouzeix, J.P., Ferland, J.A., Schaible, S.: A note on an algorithm for generalized fractional programs. J. Optim. Theory Appl. 50, 183–187 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dinkelbach, W.: On nonlinear fractional programming. Manage. Sci. 13(2), 492–498 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  13. Frenk, J.B.G., Schaible, S.: Fractional Programming. ERIM Report Series, Reference No. ERS-2004-074-LIS (2004)

  14. Gugat, M.: Prox-regularization methods for generalized fractional programming. J. Optim. Theory Appl. 99(3), 691–722 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)

    MATH  Google Scholar 

  16. Jagannathan, R., Schaible, S.: Duality in generalized fractional programming via Farkas lemma. J. Optim. Theory Appl. 41(3), 417–424 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin, J.-Y., Chen, H.-J., Sheu, R.-L.: Augmented lagrange primal-dual approach for generalized fractional programming problems. J. Ind. Manag. Opt. 4(9), 723–741 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Nagih, A., Plateau, G.: Problèmes fractionnaires: tour d’horizon sur les applications et méthodes de résolution. RAIRO Oper. Res. 33(4), 383–419 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  20. Roubi, A.: Method of centers for generalized fractional programming. J. Optim. Theory Appl. 107(1), 123–143 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Roubi, A.: Some properties of methods of centers. Comput. Opt. Appl. 19, 319–335 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Roubi, A.: Convergence of prox-regularization methods for generalized fractional programming. RAIRO Oper. Res. 36(1), 73–94 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schaible, S.: Bibliography in fractional programming. Zeitschrift für Oper. Res. 26(1), 211–241 (1982)

    MathSciNet  MATH  Google Scholar 

  24. Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P.M. (eds.) Handbook Global Optimization, pp. 495–608. Kluwer, Dordrecht (1995)

    Chapter  Google Scholar 

  25. Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stancu-Minasian, I.M.: A seventh bibliography of fractional programming. Adv. Model. Opt. 15(2), 309–386 (2013)

    Google Scholar 

  27. Strodiot, J.J., Crouzeix, J.P., Nguyen, V.H., Ferland, J.A.: An inexact proximal point method for solving generalized fractional programs. J. Global Optim. 42(1), 121–138 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculté des Sciences et Techniques, Settat, Morocco

    K. Boufi & A. Roubi

Authors
  1. K. Boufi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Roubi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Roubi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boufi, K., Roubi, A. Dual method of centers for solving generalized fractional programs. J Glob Optim 69, 387–426 (2017). https://doi.org/10.1007/s10898-017-0523-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-017-0523-z

Keywords

Mathematics Subject Classification

Search

Navigation