Skip to main content
SpringerLink
Log in

Solving the sum-of-ratios problem by a stochastic search algorithm

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In spite of the recent progress in fractional programming, the sum-of-ratios problem remains untoward. Freund and Jarre proved that this is an NP-complete problem. Most methods overcome the difficulty using the deterministic type of algorithms, particularly, the branch-and-bound method. In this paper, we propose a new approach by applying the stochastic search algorithm introduced by Birbil, Fang and Sheu to a transformed image space. The algorithm then computes and moves sample particles in the q − 1 dimensional image space according to randomly controlled interacting electromagnetic forces. Numerical experiments on problems up to sum of eight linear ratios with a thousand variables are reported. The results also show that solving the sum-of-ratios problem in the image space as proposed is, in general, preferable to solving it directly in the primal domain.

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. Almogy Y., Levin O.: A class of fractional programming problems. Oper. Res. 19, 57–67 (1971)

    Article  Google Scholar 

  2. Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity. Plenum Press (1988)

  3. Barros A.I., Frenk J.B.G., Schaible S., Zhang S.: A new algorithm for generalized fractional programs. Math. Program. 72, 147–175 (1996)

    Google Scholar 

  4. Benson H.P.: Global optimization of nonlinear sums of ratios. J. Math. Anal. Appl. 263, 301–315 (2001)

    Article  Google Scholar 

  5. Benson H.P.: Global optimization algorithm for the nonlinear sum of ratios problem. J. Optim. Theor. Appl. 112, 1–29 (2002)

    Article  Google Scholar 

  6. Benson H.P.: Using concave envelopes to globally solve the nonlinear sum of ratios problems. J. Glob. Optim. 22, 343–364 (2002)

    Article  Google Scholar 

  7. Benson H.P.: On the global optimization of sums of linear fractional functions over a convex set. J. Optim. Theor. Appl. 121, 19–39 (2004)

    Article  Google Scholar 

  8. Birbil Ş.İ., Fang S.C.: An electromagnetism-like mechanism for global optimization. J. Glob. Optim. 25, 263–282 (2003)

    Article  Google Scholar 

  9. Birbil Ş.İ., Fang S.C., Sheu R.L.: On the convergence of a population-based global optimization algorithm. J. Glob. Optim. 30, 301–318 (2004)

    Article  Google Scholar 

  10. Cambini A., Martein L., Schaible S.: On maximizing a sum of ratios. J. Inform. Optim. Sci. 10, 65–79 (1989)

    Google Scholar 

  11. Frenk J.B.G., Schaible S.: Fractional programming. In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization, vol. II, pp. 162–172. Kluwer Publishers, Dordrecht (1989)

    Google Scholar 

  12. Freund R.W., Jarre F.: Solving the sum-of-ratios problem by an interior-point method. J. Glob. Optim. 19, 83–102 (2001)

    Article  Google Scholar 

  13. Ibaraki T.: Parametric approaches to fractional programs. Math. Program. 26, 345–362 (1983)

    Article  Google Scholar 

  14. Ingber L.: Simulated annealing: practice versus theory. Math. Comput. Modell. 18, 29–57 (1993)

    Article  Google Scholar 

  15. Konno H., Abe N.: Minimization of the sum of three linear fractional functions. J. Glob. Optim. 15, 419–432 (1999)

    Article  Google Scholar 

  16. Konno H., Fukaishi K.: A branch and bound algorithm for solving low rank linear multiplicative and fractional programming problems. J. Glob. Optim. 18, 283–299 (2000)

    Article  Google Scholar 

  17. Konno H., Inori M.: Bond portfolio optimization by bilinear fractional programming. J. Oper. Res. Soc. Jpn. 32, 143–158 (1989)

    Google Scholar 

  18. Kuno T.: A branch-and-bound algorithm for maximizing the sum of several linear ratios. J. Glob. Optim. 22, 155–174 (2002)

    Article  Google Scholar 

  19. Kuno T., Yajima Y., Konno H.: An outer approximation method for minimizing the product of several convex functions on a convex set. J. Glob. Optim. 3, 325–335 (1993)

    Article  Google Scholar 

  20. Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag xvi+340 (1994)

  21. Phuong N.T.H., Tuy H.: A unified monotonic approach to generalized linear fractional programming. J. Glob. Optim. 26, 229–259 (2003)

    Article  Google Scholar 

  22. Rinnooy-Kan A.H.G., Timmer G.T.: Stochastic global optimization methods. II. Multilevel methods. Math. Program. 39, 57–78 (1987)

    Article  Google Scholar 

  23. Schaible S.: Recent results in fractional programming. Oper. Res. Verfahren 23, 271–272 (1977)

    Google Scholar 

  24. Wood G.R.: Multidimensional bisection applied to global optimisation. Comput. Math. Appl. 21, 161–172 (1991)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Wei-Ying Wu

  2. Department of Mathematics, National Cheng Kung University, Tainan, 701, Taiwan

    Ruey-Lin Sheu

  3. Faculty of Engineering and Natural Sciences, Sabancı University, Orhanli-Tuzla, 34956, Istanbul, Turkey

    Ş. İlker Birbil

Authors
  1. Wei-Ying Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruey-Lin Sheu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ş. İlker Birbil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruey-Lin Sheu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, WY., Sheu, RL. & Birbil, Ş.İ. Solving the sum-of-ratios problem by a stochastic search algorithm. J Glob Optim 42, 91–109 (2008). https://doi.org/10.1007/s10898-008-9285-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-008-9285-y

Keywords

Search

Navigation