The purpose of this paper is threefold; (1) Offer a synopsis of algorithmic review and to make a comparison between two branch-and-bound approaches for solving the sum-of-ratios problem; (2) Modify an promising algorithm for nonlinear sum-of-ratios problem; (3) Study the efficiency of the algorithms via numerical experiments.


Sum-of-ratios problem fractional programming branch-and-bound approach 


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  1. [Be02]
    Benson, H.P.: Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22, 343–364 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [Ku02]
    Kuno, T.: A branch-and-bound algorithm for maximizing the sum of several linear ratios. Journal of Global Optimization 22, 155–174 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [Cr88]
    Craven, B.D.: Fractional Programming. Sigma Series in Applied Mathematics, vol. 4. Heldermann verlag, Berlin (1988)zbMATHGoogle Scholar
  4. [FrJa01]
    Freund, R.W., Jarre, F.: Solving the sum-of-ratios problem by an interior-point method. Journal of Global Optimization 19, 83–102 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [HoTu03]
    Hoai Phuong, N.T., Tuy, H.: A unified monotonic approach to generalized linear fractional programming. Journal of Global Optimization 26, 229–259 (2003)zbMATHCrossRefGoogle Scholar
  6. [KoAb99]
    Konno, H., Abe, N.: Minimization of the sum of three linear fractional functions. Journal of Global Optimization 15, 419–432 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [KoFu00]
    Konno, H., Fukaishi, K.: A branch-and-bound algorithm for solving low rank linear multiplicative and fractional programming problems. Journal of Global Optimization 18, 283–299 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [KoYaMa91]
    Konno, H., Yajima, Y., Matsui, T.: Parametric simplex algorithms for solving a special class of nonconvex minimization problems. Journal of Global Optimization 1, 65–81 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [KoYa99]
    Konno, H., Yamashita, H.: Minimization of the sum and the product of several linear fractional functions. Naval Research Logistics 46, 583–596 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [MuTaSc95]
    Muu, L.D., Tam, B.T., Schaible, S.: Efficient algorithms for solving certain nonconvex programs dealing with the product of two affine fractional functions. Journal of Global Optimization 6, 179–191 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Ma01]
    Mastumi, T.: NP Hardness of Linear Multiplicative Programming And Related Problems. Journal of Information and Optimization Sciences 9, 113–119 (1996)Google Scholar
  12. [Sc77]
    Schaible, S.: A note on the sum of a linear and linear-fractional function. Naval Research Logistics Quarterly 24, 691–693 (1977)zbMATHCrossRefGoogle Scholar
  13. [ScSh03]
    Schaible, S., Shi, J.: Fractional programming: the sum-of-ratios case. Optimization Methods and Software 18, 219–229 (2003)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Lianbo Gao
    • 1
  • Jianming Shi
    • 1
  1. 1.Department of Computer Science and Systems EngineeringMuroran Institute of Technology 

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