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Proximal Bundle Algorithms for Nonlinearly Constrained Convex Minimax Fractional Programs

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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.

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References

  1. Falk, J.: Maximization of signal-to-noise ratio in an optical filter. SIAM J. Appl. Math. 17, 582–592 (1969)

    Article  MathSciNet  MATH  Google Scholar 

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

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. 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 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Boufi, K., Roubi, A.: Dual method of centers for solving generalized fractional programs. J. Glob. Optim. 69(2), 387–426 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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 

  21. El Haffari, M., Roubi, A.: Prox-dual regularization algorithm for generalized fractional programs. J. Ind. Manag. Optim. 13(4), 1991–2013 (2017)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Addoune, S., El Haffari, M., Roubi, A.: A proximal point algorithm for generalized fractional programs. Optimization 66(9), 1495–1517 (2017)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  25. Moreau, J.J.: Proximité et Dualité dans un Espace Hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  26. Martinet, B.: Régularisation d’inéquations variationnelles par approximation successives. Rev. Fr. d’Inf. Rech. Opér. 4, 154–158 (1970)

    MATH  Google Scholar 

  27. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  28. Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29(2), 403–419 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lemaréchal, C.: Bundle methods in nonsmooth optimization. In: Lemaréchal, C., Mifflin, R. (eds.) Nonsmooth Optimization. Pergamon Press, Oxford (1978)

    Google Scholar 

  30. Kiwiel, K.C.: An aggregate subgradient method for nonsmooth convex minimization. Math. Program. 27, 320–341 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  31. Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62, 261–275 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  33. Huard, P.: Programmation mathématique convexe. R.I.R.O. 7, 43–59 (1968)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Auslender, A.: Numerical methods for nondifferentiable convex optimization. Math. Program. Stud. 30, 102–126 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mifflin, R.: An algorithm for constrained optimization with semismooth functions. Math. Oper. Res. 2, 191–207 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mifflin, R.: A modification and extension of Lemaréchal’s algorithm for nonsmooth minimization. Math. Program. Stud. 17, 77–90 (1982)

    Article  MATH  Google Scholar 

  38. Fukushima, M.: A descent algorithm for nonsmooth convex optimization. Math. Program. 30, 163–175 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  40. Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  41. Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46, 105–122 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mäkelä, M.: Survey of bundle methods for nonsmooth optimization. Optim. Methods Softw. 17(1), 1–29 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. Polyak, B.T.: Introduction to Optimization. Translations Series in Mathematics and Engineering. Optimization Software, Inc. Publications Division, New York (1987)

  45. Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31(5), 1340–1359 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sion, M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  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)

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Acknowledgements

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

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Authors and Affiliations

  1. Faculté MI, Département de Mathématiques, University of Bordj Bou Arréridj, Bordj Bou Arréridj, Algeria

    Smail Addoune

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

    Karima Boufi & Ahmed Roubi

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  1. Smail Addoune
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  2. Karima Boufi
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  3. Ahmed Roubi
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Correspondence to Ahmed Roubi.

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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

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  • DOI: https://doi.org/10.1007/s10957-018-1342-1

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