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Branch-and-Bound Outer Approximation Algorithm for Sum-of-Ratios Fractional Programs

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  • H. P. Benson

    (University of Florida)

Abstract

In this article, a branch and-bound outer approximation algorithm is presented for globally solving a sum-of-ratios fractional programming problem. To solve this problem, the algorithm instead solves an equivalent problem that involves minimizing an indefinite quadratic function over a nonempty, compact convex set. This problem is globally solved by a branch-and-bound outer approximation approach that can create several closed-form linear inequality cuts per iteration. In contrast to pure outer approximation techniques, the algorithm does not require computing the new vertices that are created as these cuts are added. Computationally, the main work of the algorithm involves solving a sequence of convex programming problems whose feasible regions are identical to one another except for certain linear constraints. As a result, to solve these problems, an optimal solution to one problem can potentially be used to good effect as a starting solution for the next problem.

Suggested Citation

  • H. P. Benson, 2010. "Branch-and-Bound Outer Approximation Algorithm for Sum-of-Ratios Fractional Programs," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 1-18, July.
    • Handle: RePEc:spr:joptap:v:146:y:2010:i:1:d:10.1007_s10957-010-9647-8
    DOI: 10.1007/s10957-010-9647-8
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    References listed on IDEAS

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    1. Arthur F. Veinott, 1967. "The Supporting Hyperplane Method for Unimodal Programming," Operations Research, INFORMS, vol. 15(1), pages 147-152, February.
    2. 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.
    3. H. P. Benson, 2004. "On the Global Optimization of Sums of Linear Fractional Functions over a Convex Set," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 19-39, April.
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    Cited by:

    1. M. N. Yarahmadi & S. A. MirHassani & F. Hooshmand, 2023. "A heuristic method to find a quick feasible solution based on the ratio programming," Operational Research, Springer, vol. 23(3), pages 1-19, September.
    2. Yaohua Hu & Carisa Kwok Wai Yu & Xiaoqi Yang, 2019. "Incremental quasi-subgradient methods for minimizing the sum of quasi-convex functions," Journal of Global Optimization, Springer, vol. 75(4), pages 1003-1028, December.
    3. YongJin Kim & YunChol Jong & JinWon Yu, 2021. "A parametric solution method for a generalized fractional programming problem," Indian Journal of Pure and Applied Mathematics, Springer, vol. 52(4), pages 971-989, December.
    4. Mahdiloo, Mahdi & Toloo, Mehdi & Duong, Thach-Thao & Farzipoor Saen, Reza & Tatham, Peter, 2018. "Integrated data envelopment analysis: Linear vs. nonlinear model," European Journal of Operational Research, Elsevier, vol. 268(1), pages 255-267.
    5. Huang, Bingdi & Shen, Peiping, 2024. "An efficient branch and bound reduction algorithm for globally solving linear fractional programming problems," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).

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