Journal of Global Optimization

, Volume 26, Issue 3, pp 229–259 | Cite as

A Unified Monotonic Approach to Generalized Linear Fractional Programming

  • Nguyen Thi Hoai Phuong
  • Hoang Tuy


We present an efficient unified method for solving a wide class of generalized linear fractional programming problems. This class includes such problems as: optimizing (minimizing or maximizing) a pointwise maximum or pointwise minimum of a finite number of ratios of linear functions, optimizing a sum or product of such ratios, etc. – over a polytope. Our approach is based on the recently developed theory of monotonic optimization.

Generalized fractional programming Sum or product of ratios of linear functions Monotonic optimization Global optimization Polyblock approximation approach 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barros, A.I. (1998), Discrete and Fractional Programming Techniques for Loaction Models, Kluwer.Google Scholar
  2. 2.
    Barros, A.I. and Frenk, J.B.G. (1995), Generalized fractional programming and cutting plane algorithms', Journal of Optimization Theory and Applications 87: 103–120.Google Scholar
  3. 3.
    Benson, H.P. and Boger, G.M. (2000), Outcome-space cutting plane algorithm for linear multiplicative programming, Journal of Optimization Theory and Applications 104: 301–322.Google Scholar
  4. 4.
    Cambini, A., Martein, L. and Schaible, S. (1989), On maximizing a sum of ratios, Journal of Information and Optimization Science 10: 141–151.Google Scholar
  5. 5.
    Charnes, A. and Cooper, W.W. (1962), Programming with Linear Fractional Functionals, Naval Research Logistics Quaterly 9: 181–186.Google Scholar
  6. 6.
    Craven, B.D. (1988), Fractional Programming, Heldermann Verlag, Berlin.Google Scholar
  7. 7.
    Falk, J.E. and Palocsay, S.W. (1992), Optimizing the sum of linear fractional functions. In: Floudas, C. and Pardalos, P. (eds.), Global Optimization. Princeton University Press, pp. 221–258.Google Scholar
  8. 8.
    Crouzeix, J.P., Ferland, J.A. and Schaible, S. (1985), An algorithm for generalized fractional programming, Journal of Optimization Theory and Applications 47: 35–49.Google Scholar
  9. 9.
    Konno, H. and Abe, N. (1999), Minimization of the sum of three linear fractional functions, Journal of Global Optimization 15: 419–432.Google Scholar
  10. 10.
    Konno, H. and Kuno, T. (1992), Linear multiplicative programming, Mathematical Programming 56: 51–64.Google Scholar
  11. 11.
    Konno, H., Thach, P.T. and Tuy, H. (1997), Optimization on Low Rank Nonconvex Structures, Kluwer Academic Publishers, Dordrecht/Boston/London.Google Scholar
  12. 12.
    Konno, H., Yajima, Y. and Matsui, T. (1991), Parametric simplex algorithms for solving a special class of nonconvex minimization problems, Journal of Global Optimization 1: 65–81.Google Scholar
  13. 13.
    Konno, H. and Yajima, Y. (1992), Minimizing and maximizing the product of linear fractional functions. In: Floudas, C. and Pardalos, P. (eds.), Recent Advances in Global Optimization eds. Princeton University Press, pp. 259–273.Google Scholar
  14. 14.
    Konno, H. and Yamashita, Y. (1997), Minimization of the sum and the product of several linear fractional functions, Tokyo Institute of Technology, Technical Report, Department of IE & Management, to appear in Naval Research Logistics.Google Scholar
  15. 15.
    Luc, L.T. (2001), Reverse polyblock approximation for optimization over the weakly efficient set and efficient set, Acta Mathematica Vietnamica 26: 65–80.Google Scholar
  16. 16.
    Rubinov, A., Tuy, H. and Mays, H. (2001), Algorithm for a monotonic global optimization problem, Optimization 49: 205–221.Google Scholar
  17. 17.
    Stancu-Minasian, I.M. (1997), Fractional Programming: Theory, Methods and Applications, Kluwer, Dordrecht.Google Scholar
  18. 18.
    Schaible, S. (1992), Fractional Programming. In: Horst, R. and Pardalos, P. (eds.), Handbook of Global Optimization, Kluwer, Dordrecht, pp. 495–608.Google Scholar
  19. 19.
    Tawarmalani, M. and Sahinidis, N.V. (2001), Semiddefinite relaxations of fractional programs via novel convexification techniques, Journal of Global Optimization 20: 137–158.Google Scholar
  20. 20.
    Tuy, H. (1998), Convex Analysis and Global Optimization, Kluwer, Dordrecht.Google Scholar
  21. 21.
    Tuy, H. (1999), Normal sets, polyblocks and monotonic optimization, Vietnam Journal of Mathematics 27(4): 277–300.Google Scholar
  22. 22.
    Tuy, H. (2000), Monotonic optimization: problems and solution approaches, SIAM Journal of Optimization 11(2): 464–494.Google Scholar
  23. 23.
    Tuy, H. and Luc, L.T. (2000), A new approach to optimization under monotonic constraint, Journal of Global Optimization 18: 1–15.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Nguyen Thi Hoai Phuong
    • 1
  • Hoang Tuy
    • 1
  1. 1.Institute of MathematicsBo Ho, HanoiVietnam

Personalised recommendations