Advertisement

Journal of Optimization Theory and Applications

, Volume 179, Issue 1, pp 212–239 | Cite as

Proximal Bundle Algorithms for Nonlinearly Constrained Convex Minimax Fractional Programs

  • Smail Addoune
  • Karima Boufi
  • Ahmed Roubi
Article
  • 96 Downloads

Abstract

A generalized fractional programming problem is defined as the problem of minimizing a nonlinear function, defined as the maximum of several ratios of functions on a feasible domain. In this paper, we propose new methods based on the method of centers, on the proximal point algorithm and on the idea of bundle methods, for solving such problems. First, we introduce proximal point algorithms, in which, at each iteration, an approximate prox-regularized parametric subproblem is solved inexactly to obtain an approximate solution to the original problem. Based on this approach and on the idea of bundle methods, we propose implementable proximal bundle algorithms, in which the objective function of the last mentioned prox-regularized parametric subproblem is replaced by an easier one, typically a piecewise linear function. The methods deal with nondifferentiable nonlinearly constrained convex minimax fractional problems. We prove the convergence, give the rate of convergence of the proposed procedures and present numerical tests to illustrate their behavior.

Keywords

Generalized fractional programs Method of centers Proximal point algorithm Bundle methods Quadratic programming 

Mathematics Subject Classification

90C32 90C25 49K35 49M37 

Notes

Acknowledgements

The authors would like to thank an anonymous referee for his careful reading of the manuscript and his valuable remarks.

References

  1. 1.
    Falk, J.: Maximization of signal-to-noise ratio in an optical filter. SIAM J. Appl. Math. 17, 582–592 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P.M. (eds.) Handbook Global Optimization, pp. 495–608. Kluwer, Dordrecht (1995)CrossRefGoogle Scholar
  3. 3.
    Nagih, A., Plateau, G.: Problèmes fractionnaires: tour d’horizon sur les applications et méthodes de résolution. RAIRO-Oper. Res. 33(4), 383–419 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Frenk, J.B.G., Schaible, S.: Fractional programming. ERIM Report Series. Reference No. ERS-2004-074-LIS (2004)Google Scholar
  5. 5.
    Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13(2), 492–498 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crouzeix, J.P., Ferland, J.A., Schaible, S.: An algorithm for generalized fractional programs. J. Optim. Theory Appl. 47, 35–49 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bernard, J.C., Ferland, J.A.: Convergence of interval-type algorithms for generalized fractional programming. Math. Program. 43, 349–363 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Crouzeix, J.P., Ferland, J.A.: Algorithms for generalized fractional programming. Math. Program. 52, 191–207 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Roubi, A.: Method of centers for generalized fractional programming. J. Optim. Theory Appl. 107(1), 123–143 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jagannathan, R., Schaible, S.: Duality in generalized fractional programming via Farkas lemma. J. Optim. Theory Appl. 41(3), 417–424 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Crouzeix, J.P., Ferland, J.A., Schaible, S.: Duality in generalized linear fractional programming. Math. Program. 27, 342–354 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Barros, A.I., Frenk, J.B.G., Schaible, S., Zhang, S.: A new algorithm for generalized fractional programs. Math. Program. 72, 147–175 (1996)MathSciNetzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bector, C.R., Chandra, S., Bector, M.K.: Generalized fractional programming duality: a parametric approach. J. Optim. Theory Appl. 60(2), 243–260 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Boufi, K., Roubi, A.: Dual method of centers for solving generalized fractional programs. J. Glob. Optim. 69(2), 387–426 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gugat, M.: Prox-regularization methods for generalized fractional programming. J. Optim. Theory Appl. 99(3), 691–722 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Roubi, A.: Convergence of prox-regularization methods for generalized fractional programming. RAIRO-Oper. Res. 36(1), 73–94 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Addou, A., Roubi, A.: Proximal-type methods with generalized Bregman functions and applications to generalized fractional programming. Optimization 59(7), 1085–1105 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    El Haffari, M., Roubi, A.: Prox-dual regularization algorithm for generalized fractional programs. J. Ind. Manag. Optim. 13(4), 1991–2013 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Addoune, S., El Haffari, M., Roubi, A.: A proximal point algorithm for generalized fractional programs. Optimization 66(9), 1495–1517 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Stancu-Minasian, I.: A seventh bibliography of fractional programming. Adv. Model. Optim. 15(2), 309–386 (2013)MathSciNetGoogle Scholar
  25. 25.
    Moreau, J.J.: Proximité et Dualité dans un Espace Hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Martinet, B.: Régularisation d’inéquations variationnelles par approximation successives. Rev. Fr. d’Inf. Rech. Opér. 4, 154–158 (1970)zbMATHGoogle Scholar
  27. 27.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29(2), 403–419 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lemaréchal, C.: Bundle methods in nonsmooth optimization. In: Lemaréchal, C., Mifflin, R. (eds.) Nonsmooth Optimization. Pergamon Press, Oxford (1978)Google Scholar
  30. 30.
    Kiwiel, K.C.: An aggregate subgradient method for nonsmooth convex minimization. Math. Program. 27, 320–341 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62, 261–275 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  33. 33.
    Huard, P.: Programmation mathématique convexe. R.I.R.O. 7, 43–59 (1968)CrossRefzbMATHGoogle Scholar
  34. 34.
    Roubi, A.: Some properties of methods of centers. Comput. Optim. Appl. 19, 319–335 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Auslender, A.: Numerical methods for nondifferentiable convex optimization. Math. Program. Stud. 30, 102–126 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mifflin, R.: An algorithm for constrained optimization with semismooth functions. Math. Oper. Res. 2, 191–207 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Mifflin, R.: A modification and extension of Lemaréchal’s algorithm for nonsmooth minimization. Math. Program. Stud. 17, 77–90 (1982)CrossRefzbMATHGoogle Scholar
  38. 38.
    Fukushima, M.: A descent algorithm for nonsmooth convex optimization. Math. Program. 30, 163–175 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Lemaréchal, C.: Constructing bundle methods for convex optimization. In: Hiriart-Urruty, J.B. (ed.) Fermat Days 85: Mathematics for Optimization, pp. 201–240. North-Holland, Amsterdam (1986)CrossRefGoogle Scholar
  40. 40.
    Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  41. 41.
    Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46, 105–122 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis. Numerical results. SIAM J. Optim. 2, 121–152 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Mäkelä, M.: Survey of bundle methods for nonsmooth optimization. Optim. Methods Softw. 17(1), 1–29 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Polyak, B.T.: Introduction to Optimization. Translations Series in Mathematics and Engineering. Optimization Software, Inc. Publications Division, New York (1987)Google Scholar
  45. 45.
    Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31(5), 1340–1359 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Sion, M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Addoune, S., Boufi, K., Roubi, A.: Proximal bundle algorithms for nonlinearly constrained convex generalized fractional programs. Internal Report, Laboratoire MISI, Faculté des Sciences et Techniques, Settat (2017)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculté MI, Département de MathématiquesUniversity of Bordj Bou ArréridjBordj Bou ArréridjAlgeria
  2. 2.Laboratoire MISI, Faculté des Sciences et TechniquesUniversity Hassan 1SettatMorocco

Personalised recommendations