Skip to main content
SpringerLink
Log in

Optimality Conditions and a Method of Centers for Minimax Fractional Programs with Difference of Convex Functions

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

Abstract

We are concerned in this paper with minimax fractional programs whose objective functions are the maximum of finite ratios of difference of convex functions, with constraints also described by difference of convex functions. Like Dinkelbach-type algorithms, the method of centers for generalized fractional programs fails to work for such problems, since the parametric subproblems may be nonconvex, whereas the latters need a global optimal solution for these subproblems. We first give necessary optimality conditions for these problems, by means of convex analysis tools, and then extend the last method to solve such programs. The method is based on solving a sequence of parametric convex problems. We show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of Karush–Kuhn–Tucker criticality type.

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. Tuy, H.: Convex Analysis and Global Optimization, 1st edn. Kluwer Academic Publishers, Dordrecht (1998)

    Book  Google Scholar 

  2. Bernard, J.C., Ferland, J.A.: Convergence of interval-type algorithms for generalized fractional programming. Math. Program. 43, 349–363 (1989). https://doi.org/10.1007/BF01582298

    Article  MathSciNet  MATH  Google Scholar 

  3. Crouzeix, J.P., Ferland, J.A.: Algorithms for generalized fractional programming. Math. Program. 52, 191–207 (1991). https://doi.org/10.1007/BF01582887

    Article  MathSciNet  MATH  Google Scholar 

  4. Crouzeix, J.P., Ferland, J.A., Nguyen, H.V.: Revisiting Dinkelbach-type algorithms for generalized fractional programs. OPSEARCH 45, 97–110 (2008). https://doi.org/10.1007/BF03398807

    Article  MathSciNet  MATH  Google Scholar 

  5. Crouzeix, J.P., Ferland, J.A., Schaible, S.: An algorithm for generalized fractional programs. J. Optim. Theory Appl. 47, 35–49 (1985). https://doi.org/10.1007/BF00941314

    Article  MathSciNet  MATH  Google Scholar 

  6. 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). https://doi.org/10.1007/BF00938484

    Article  MathSciNet  MATH  Google Scholar 

  7. Roubi, A.: Method of centers for generalized fractional programming. J. Optim. Theory Appl. 107(1), 123–143 (2000). https://doi.org/10.1023/A:1004660917684

    Article  MathSciNet  MATH  Google Scholar 

  8. Roubi, A.: Convergence of prox-regularization methods for generalized fractional programming. RAIRO-Oper. Res. 36(1), 73–94 (2002). https://doi.org/10.1051/ro:2002006

    Article  MathSciNet  MATH  Google Scholar 

  9. Strodiot, J.J., Crouzeix, J.P., Ferland, J.A., Nguyen, V.H.: An inexact proximal point method for solving generalized fractional programs. J. Glob. Optim. 42(1), 121–138 (2008). https://doi.org/10.1007/s10898-007-9270-x

    Article  MathSciNet  MATH  Google Scholar 

  10. Addoune, S., Boufi, K., Roubi, A.: Proximal bundle algorithms for nonlinearly constrained convex minimax fractional programs. J. Optim. Theory Appl. 179, 212–239 (2018). https://doi.org/10.1007/s10957-018-1342-1

    Article  MathSciNet  MATH  Google Scholar 

  11. Addoune, S., El Haffari, M., Roubi, A.: A proximal point algorithm for generalized fractional programs. Optimization 66(9), 1495–1517 (2017). https://doi.org/10.1080/02331934.2017.1338698

    Article  MathSciNet  MATH  Google Scholar 

  12. Barros, A.I., Frenk, J.B.G., Schaible, S., Zhang, S.: A new algorithm for generalized fractional programs. Math. Program. 72, 147–175 (1996). https://doi.org/10.1007/BF02592087

    Article  MathSciNet  MATH  Google Scholar 

  13. Barros, A.I., Frenk, J.B.G., Schaible, S., Zhang, S.: Using duality to solve generalized fractional programming problems. J. Glob. Optim. 8, 139–170 (1996). https://doi.org/10.1007/BF00138690

    Article  MathSciNet  MATH  Google Scholar 

  14. Bector, C.R., Chandra, S., Bector, M.K.: Generalized fractional programming duality: a parametric approach. J. Optim. Theory Appl. 60(2), 243–260 (1989). https://doi.org/10.1007/BF00940006

    Article  MathSciNet  MATH  Google Scholar 

  15. Boualam, H., Roubi, A.: Dual algorithms based on the proximal bundle method for solving convex minimax fractional programs. J. Ind. Manag. Optim. 15(4), 1897–1920 (2019). https://doi.org/10.3934/jimo.2018128

    Article  MathSciNet  MATH  Google Scholar 

  16. Boualam, H., Roubi, A.: Proximal bundle methods based on approximate subgradients for solving Lagrangian duals of minimax fractional programs. J. Glob. Optim. 74(2), 255–284 (2019). https://doi.org/10.1007/s10898-019-00757-2

    Article  MathSciNet  MATH  Google Scholar 

  17. Boufi, K., Roubi, A.: Dual method of centers for solving generalized fractional programs. J. Glob. Optim. 69(2), 387–426 (2017). https://doi.org/10.1007/s10898-017-0523-z

    Article  MathSciNet  MATH  Google Scholar 

  18. Boufi, K., Roubi, A.: Duality results and dual bundle methods based on the dual method of centers for minimax fractional programs. SIAM J. Optim. 29(2), 1578–1602 (2019)

    Article  MathSciNet  Google Scholar 

  19. Boufi, K., Roubi, A.: Prox-regularization of the dual method of centers for generalized fractional programs. Optim. Methods Softw. 34(3), 515–545 (2019). https://doi.org/10.1080/10556788.2017.1392520

    Article  MathSciNet  MATH  Google Scholar 

  20. Crouzeix, J.P., Ferland, J.A., Schaible, S.: Duality in generalized linear fractional programming. Math. Program. 27, 342–354 (1983). https://doi.org/10.1007/BF02591908

    Article  MathSciNet  MATH  Google Scholar 

  21. El Haffari, M., Roubi, A.: Convergence of a proximal algorithm for solving the dual of a generalized fractional program. RAIRO-Oper. Res. 51(4), 985–1004 (2017). https://doi.org/10.1051/ro/2017004

    Article  MathSciNet  MATH  Google Scholar 

  22. El Haffari, M., Roubi, A.: Prox-dual regularization algorithm for generalized fractional programs. J. Ind. Manag. Optim. 13(4), 1991–2013 (2017). https://doi.org/10.3934/jimo.2017028

    Article  MathSciNet  MATH  Google Scholar 

  23. Jagannathan, R., Schaible, S.: Duality in generalized fractional programming via Farkas’ lemma. J. Optim. Theory Appl. 41(3), 417–424 (1983). https://doi.org/10.1007/BF00935361

    Article  MathSciNet  MATH  Google Scholar 

  24. Stancu, A.M.: Mathematical Programming with Type-I Functions. MatrixRom, Bucharest (2013)

    Google Scholar 

  25. Stancu-Minasian, I.M.: Fractional Programming: Theory, Methods and Applications. Kluwer Academic, Dordrecht (1997)

    Book  Google Scholar 

  26. Stancu-Minasian, I.M.: A sixth bibliography of fractional programming. Optimization 55, 405–428 (2006). https://doi.org/10.1080/02331930600819613

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  28. Stancu-Minasian, I.M.: An eighth bibliography of fractional programming. Optimization 66, 439–470 (2017). https://doi.org/10.1080/02331934.2016.1276179

    Article  MathSciNet  MATH  Google Scholar 

  29. Stancu-Minasian, I.M.: A ninth bibliography of fractional programming. Optimization 68(11), 2125–2169 (2019). https://doi.org/10.1080/02331934.2019.1632250

    Article  MathSciNet  MATH  Google Scholar 

  30. Le Thi, H.A., Pham Dinh, T.: DC programming and DCA: thirty years of developments. Math. Program. Ser. B 15, 137–161 (2015). https://doi.org/10.1007/s10107-018-1235-y

    Article  MATH  Google Scholar 

  31. Pham Dinh, T., El Bernoussi, S.: Algorithms for solving a class of nonconvex optimization problems. Methods of subgradients. In: Fermat Days 85: Mathematics for Optimization. North-Holland Mathematics Studies, vol. 129. North-Holland, Amsterdam (1986)

  32. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  33. Hiriart-Urruty, J.B.: Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Ponstein, J. (ed.) Convexity and Duality in Optimization, vol. 256. Springer, Berlin (1985)

  34. Le Thi, H.A., Huynh, V.N., Pham Dinh, T.: DC programming and DCA for general DC programs. In: van Do T., Thi, H., Nguyen, N. (eds.) Advanced Computational Methods for Knowledge Engineering. Advances in Intelligent Systems and Computing, vol. 282, pp. 15–35. Springer, Cham (2014). https://doi.org/10.1007/s10107-018-1235-y

  35. Le Thi, H.A., Pham Dinh, T.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46 (2005)

    Article  MathSciNet  Google Scholar 

  36. Buì-Trong-Liêũ, Huard, P.: La Méthode des Centres dans un Espace Topologique. Numerische Mathematik 8, 56–67 (1966)

  37. Huard, P.: Programmation Mathématique Convexe. R.I.R.O. 2(7), 43–59 (1968)

    MathSciNet  MATH  Google Scholar 

  38. Roubi, A.: Global convergence and rate of convergence of a methods of centers. Comput. Optim. Appl. 3, 259–280 (1994)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. Sagastizábal, C., Solodov, M.: An infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter. SIAM J. Optim. 16(1), 146–169 (2005)

    Article  MathSciNet  Google Scholar 

  41. Ackooij, W., Sagastizábal, C.: Constrained bundle methods for upper inexact oracles with application to joint chance constrained energy problems. SIAM J. Optim. 24(2), 733–765 (2014)

    Article  MathSciNet  Google Scholar 

  42. Lv, J., Pang, L., Meng, F.: A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information. J. Glob. Optim. 70(4), 517–549 (2018). https://doi.org/10.1007/s10898-017-0565-2

    Article  MathSciNet  MATH  Google Scholar 

  43. Polak, E., He, L.: Unified steerable phase I-phase II method of feasible directions for semi-infinite optimization. Math. Program. 59(1), 83–107 (1991)

    MathSciNet  MATH  Google Scholar 

  44. Polak, E., Trahan, R., Mayne, D.Q.: Combined phase I-phase II method of feasible directions. Math. Program. 17, 61–73 (1979)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the reviewers for their careful reading of the paper and thank them for their remarks, corrections and suggestions.

Author information

Authors and Affiliations

  1. Laboratoire MISI, Faculté des Sciences et Techniques, Univ. Hassan 1, Settat, Morocco

    Karima Boufi & Ahmed Roubi

  2. Laboratoire MISI & Ecole Normale Supérieure, Univ. Abdelmalek Essaâdi, Tétouan, Morocco

    Mostafa El Haffari

Authors
  1. Karima Boufi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mostafa El Haffari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahmed Roubi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed Roubi.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boufi, K., El Haffari, M. & Roubi, A. Optimality Conditions and a Method of Centers for Minimax Fractional Programs with Difference of Convex Functions. J Optim Theory Appl 187, 105–132 (2020). https://doi.org/10.1007/s10957-020-01738-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-020-01738-2

Keywords

Mathematics Subject Classification

Search

Navigation