Journal of Global Optimization

, Volume 33, Issue 2, pp 215–234 | Cite as

A Revision of the Trapezoidal Branch-and-Bound Algorithm for Linear Sum-of-Ratios Problems

  • Takahito KunoEmail author


In this paper, we point out a theoretical flaw in Kuno [(2002)Journal of Global Optimization 22, 155–174] which deals with the linear sum-of-ratios problem, and show that the proposed branch-and-bound algorithm works correctly despite the flaw. We also note a relationship between a single ratio and the overestimator used in the bounding operation, and develop a procedure for tightening the upper bound on the optimal value. The procedure is not expensive, but the revised algorithms incorporating it improve significantly in efficiency. This is confirmed by numerical comparisons between the original and revised algorithms.


Branch-and-bound algorithm Fractional programming Global optimization Nonconvex optimization Sum-of-ratios problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Almogy, Y., Levin, O. 1970Parametric analysis of a multi-stage stochastic shipping problemLawrence, J. eds. Operational Research ’69Tavistock PublicationsLondon359370Google Scholar
  2. 2.
    Avriel, M., Diewert, W.E., Schaible, S., Zang, I. 1988Generalized ConvexityPlenum PressNew YorkGoogle Scholar
  3. 3.
    Benson, H.P. 2002Using concave envelopes to globally solve the nonlinear sum of ratios problemJournal of Global Optimization22343364CrossRefGoogle Scholar
  4. 4.
    Chvátal, V. 1983Linear ProgrammingFreemanNew YorkGoogle Scholar
  5. 5.
    Crouzeix, J.P., Ferland, J.A., Schaible, S. 1985An algorithm for generalized fractional programsJournal of Optimization Theory and Applications473549CrossRefGoogle Scholar
  6. 6.
    Dür, R. Horst and Thoai, N.V. Solving sum-of-ratios fractional programs using efficient points,Optimization 49, 447–466.Google Scholar
  7. 7.
    Falk, J.E., Palocsay, S.W. 1994Image space analysis of generalized fractional programsJournal of Global Optimization46388CrossRefGoogle Scholar
  8. 8.
    Freund, R.W., Jarre, F. 2001Solving the sum-of-ratios problem by an interior-point methodJournal of Global Optimization1983102CrossRefGoogle Scholar
  9. 9.
    HoaiPhuong, N.T., Tuy, H. 2003A unified monotonic approach to generalized linear fractional programmingJournal of Global Optimization26229259CrossRefGoogle Scholar
  10. 10.
    Horst, R., Tuy, H. 1993Global Optimization: Deterministic Approaches2Springer-VerlagBerlinGoogle Scholar
  11. 11.
    Konno, H., Abe, N. 1999Minimization of the sum of three linear fractional functionsJournal of Global Optimization15419432CrossRefGoogle Scholar
  12. 12.
    Konno, H. and Fukaishi, K. A branch-and-bound algorithm for solving low rank linear multiplicative and fractional programming problems.Google Scholar
  13. 13.
    Konno, H., Thach, P.T., Tuy, H. 1997Optimization on Low Rank Nonconvex StructuresKluwer Academic PublishersDordrechtGoogle Scholar
  14. 14.
    Konno, H., Watanabe, H. 1996Bond portfolio optimization problems and their applications to index trackingJournal of the Operations Research Society of Japan39295306Google Scholar
  15. 15.
    Konno, H., Yajima, Y., Matsui, T. 1991Parametric simplex algorithms for solving a special class of nonconvex minimization problemsJournal of Global Optimization16581CrossRefGoogle Scholar
  16. 16.
    Konno, H., Yamashita, H. 1999Minimization of the sum and the product of several linear fractional functionsNaval Research Logistics46583596CrossRefGoogle Scholar
  17. 17.
    Kuno, T. 2002A branch-and-bound algorithms for maximizing the sum of several linear fractional functionsJournal of Global Optimization22155174CrossRefGoogle Scholar
  18. 18.
    Majihi, J., Janardan, R., Smid, M., Gupta, P. 1999On some geometric optimization problems in layered manufacturingComputational Geometry12219239CrossRefGoogle Scholar
  19. 19.
    Muu, L.D., Tam, B.T., Schaible, S. 1995Efficient algorithms for solving certain nonconvex programs dealing with the product of two affine fractional functionsJournal of Global Optimization6179191CrossRefGoogle Scholar
  20. 20.
    Octave Home Page,
  21. 21.
    Schaible, S. 1995Fractional programmingHorst, R.Pardalos, P.M. eds. Handbook of Global OptimizationKluwer Academic PublishersDordrecht495608Google Scholar
  22. 22.
    Schaible, S., Shi, J. 2003Fractional programming: the sum-of-ratios caseOptimization Methods and Software18219229CrossRefGoogle Scholar
  23. 23.
    Schwerdt, J., Smid, M., Janardan, R., Johnson, E. and Majihi, J. Protecting critical facets in layered manufacturing, Computational Geometry 16, 187–210.Google Scholar
  24. 24.
    Tuy, H. 2000Monotonic optimization: problems and solution approachesSIAM Journal of Optimization11464494CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Graduate School of Systems and Information EngineeringUniversity of TsukubaTsukubaJapan

Personalised recommendations