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On solving the sum-of-ratios problem

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  • Gruzdeva, Tatiana V.
  • Strekalovsky, Alexander S.

Abstract

This paper addresses the development of efficient global search methods for fractional programming problems. Such problems are, in general, nonconvex (with numerous local extremums) and belong to a class of global optimization problems. First, we reduce a rather general fractional programming problem with d.c. functions to solving an equation with a vector parameter that satisfies some nonnegativity assumption. This theorem allows the justified use of the generalized Dinkelbach’s approach for solving fractional programming problems with a d.c. goal function. Based on solving of some d.c. minimization problem, we developed a global search algorithm for fractional programming problems, which was tested on a set of low-dimensional test problems taken from the literature as well as on randomly generated problems with up to 200 variables or 200 terms in the sum.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:318:y:2018:i:c:p:260-269
    DOI: 10.1016/j.amc.2017.07.074
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    References listed on IDEAS

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    1. Alexander S. Strekalovsky, 2017. "Global Optimality Conditions in Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 770-792, June.
    2. Pandey, Pooja & Punnen, Abraham P., 2007. "A simplex algorithm for piecewise-linear fractional programming problems," European Journal of Operational Research, Elsevier, vol. 178(2), pages 343-358, April.
    3. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    4. 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.
    5. 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.
    6. Werner Dinkelbach, 1967. "On Nonlinear Fractional Programming," Management Science, INFORMS, vol. 13(7), pages 492-498, March.
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    Cited by:

    1. 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.
    2. Dey, Shibshankar & Kim, Cheolmin & Mehrotra, Sanjay, 2024. "An algorithm for stochastic convex-concave fractional programs with applications to production efficiency and equitable resource allocation," European Journal of Operational Research, Elsevier, vol. 315(3), pages 980-990.

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