Journal of Optimization Theory and Applications

, Volume 179, Issue 1, pp 212–239

Proximal Bundle Algorithms for Nonlinearly Constrained Convex Minimax Fractional Programs

• Karima Boufi
• Ahmed Roubi
Article

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)
2. 2.
Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P.M. (eds.) Handbook Global Optimization, pp. 495–608. Kluwer, Dordrecht (1995)
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)
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)
6. 6.
Crouzeix, J.P., Ferland, J.A., Schaible, S.: An algorithm for generalized fractional programs. J. Optim. Theory Appl. 47, 35–49 (1985)
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)
8. 8.
Bernard, J.C., Ferland, J.A.: Convergence of interval-type algorithms for generalized fractional programming. Math. Program. 43, 349–363 (1989)
9. 9.
Crouzeix, J.P., Ferland, J.A.: Algorithms for generalized fractional programming. Math. Program. 52, 191–207 (1991)
10. 10.
Roubi, A.: Method of centers for generalized fractional programming. J. Optim. Theory Appl. 107(1), 123–143 (2000)
11. 11.
Jagannathan, R., Schaible, S.: Duality in generalized fractional programming via Farkas lemma. J. Optim. Theory Appl. 41(3), 417–424 (1983)
12. 12.
Crouzeix, J.P., Ferland, J.A., Schaible, S.: Duality in generalized linear fractional programming. Math. Program. 27, 342–354 (1983)
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)
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)
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)
16. 16.
Boufi, K., Roubi, A.: Dual method of centers for solving generalized fractional programs. J. Glob. Optim. 69(2), 387–426 (2017)
17. 17.
Gugat, M.: Prox-regularization methods for generalized fractional programming. J. Optim. Theory Appl. 99(3), 691–722 (1998)
18. 18.
Roubi, A.: Convergence of prox-regularization methods for generalized fractional programming. RAIRO-Oper. Res. 36(1), 73–94 (2002)
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)
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)
21. 21.
El Haffari, M., Roubi, A.: Prox-dual regularization algorithm for generalized fractional programs. J. Ind. Manag. Optim. 13(4), 1991–2013 (2017)
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)
23. 23.
Addoune, S., El Haffari, M., Roubi, A.: A proximal point algorithm for generalized fractional programs. Optimization 66(9), 1495–1517 (2017)
24. 24.
Stancu-Minasian, I.: A seventh bibliography of fractional programming. Adv. Model. Optim. 15(2), 309–386 (2013)
25. 25.
Moreau, J.J.: Proximité et Dualité dans un Espace Hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)
26. 26.
Martinet, B.: Régularisation d’inéquations variationnelles par approximation successives. Rev. Fr. d’Inf. Rech. Opér. 4, 154–158 (1970)
27. 27.
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
28. 28.
Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29(2), 403–419 (1991)
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)
31. 31.
Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62, 261–275 (1993)
32. 32.
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II. Springer, Berlin (1993)
33. 33.
Huard, P.: Programmation mathématique convexe. R.I.R.O. 7, 43–59 (1968)
34. 34.
Roubi, A.: Some properties of methods of centers. Comput. Optim. Appl. 19, 319–335 (2001)
35. 35.
Auslender, A.: Numerical methods for nondifferentiable convex optimization. Math. Program. Stud. 30, 102–126 (1987)
36. 36.
Mifflin, R.: An algorithm for constrained optimization with semismooth functions. Math. Oper. Res. 2, 191–207 (1977)
37. 37.
Mifflin, R.: A modification and extension of Lemaréchal’s algorithm for nonsmooth minimization. Math. Program. Stud. 17, 77–90 (1982)
38. 38.
Fukushima, M.: A descent algorithm for nonsmooth convex optimization. Math. Program. 30, 163–175 (1984)
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)
40. 40.
Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics. Springer, Berlin (1985)
41. 41.
Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46, 105–122 (1990)
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)
43. 43.
Mäkelä, M.: Survey of bundle methods for nonsmooth optimization. Optim. Methods Softw. 17(1), 1–29 (2002)
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)
46. 46.
Sion, M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)
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

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