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A branch-and-cut algorithm for a class of sum-of-ratios problems

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  • Ashtiani, Alireza M.
  • Ferreira, Paulo A.V.

Abstract

The problem of maximizing a sum of concave–convex ratios over a convex set is addressed. The projection of the problem onto the image space of the functions that describe the ratios leads to the equivalent problem of maximizing a sum of elementary ratios subject to a linear semi-infinite inequality constraint. A global optimization algorithm that integrates a branch-and-bound procedure for dealing with nonconcavities in the image space and an efficient relaxation procedure for handling the semi-infinite constraint is proposed and illustrated through numerical examples. Comparative (computational) analyses between the proposed algorithm and two alternative algorithms for solving sum-of-ratios problems are also presented.

Suggested Citation

  • Ashtiani, Alireza M. & Ferreira, Paulo A.V., 2015. "A branch-and-cut algorithm for a class of sum-of-ratios problems," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 596-608.
    • Handle: RePEc:eee:apmaco:v:268:y:2015:i:c:p:596-608
    DOI: 10.1016/j.amc.2015.06.089
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    References listed on IDEAS

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    1. Alireza M. Ashtiani & Paulo A. V. Ferreira, 2011. "On the Solution of Generalized Multiplicative Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 411-419, May.
    2. Cambini, Riccardo & Sodini, Claudio, 2010. "A unifying approach to solve some classes of rank-three multiplicative and fractional programs involving linear functions," European Journal of Operational Research, Elsevier, vol. 207(1), pages 25-29, November.
    3. Benson, Harold P., 2007. "A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem," European Journal of Operational Research, Elsevier, vol. 182(2), pages 597-611, October.
    4. Arthur M. Geoffrion, 1970. "Elements of Large-Scale Mathematical Programming Part I: Concepts," Management Science, INFORMS, vol. 16(11), pages 652-675, July.
    5. H. P. Benson, 2002. "Global Optimization Algorithm for the Nonlinear Sum of Ratios Problem," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 1-29, January.
    6. Rúbia Oliveira & Paulo Ferreira, 2010. "An outcome space approach for generalized convex multiplicative programs," Journal of Global Optimization, Springer, vol. 47(1), pages 107-118, May.
    7. Schaible, Siegfried, 1981. "Fractional programming: Applications and algorithms," European Journal of Operational Research, Elsevier, vol. 7(2), pages 111-120, June.
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    Cited by:

    1. Bo Zhang & YueLin Gao & Xia Liu & XiaoLi Huang, 2022. "An Outcome-Space-Based Branch-and-Bound Algorithm for a Class of Sum-of-Fractions Problems," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 830-855, March.
    2. Jiao, Hongwei & Ma, Junqiao, 2022. "An efficient algorithm and complexity result for solving the sum of general affine ratios problem," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    3. Shen, Peiping & Zhu, Zeyi & Chen, Xiao, 2019. "A practicable contraction approach for the sum of the generalized polynomial ratios problem," European Journal of Operational Research, Elsevier, vol. 278(1), pages 36-48.
    4. Gruzdeva, Tatiana V. & Strekalovsky, Alexander S., 2018. "On solving the sum-of-ratios problem," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 260-269.
    5. Bo Zhang & YueLin Gao & Xia Liu & XiaoLi Huang, 2023. "Outcome-space branch-and-bound outer approximation algorithm for a class of non-convex quadratic programming problems," Journal of Global Optimization, Springer, vol. 86(1), pages 61-92, May.

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