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An efficient global optimization algorithm for the sum of linear ratios problems based on a novel adjustable branching rule

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  • Bingdi Huang

    (Henan Normal University
    Henan Institute of Science and Technology)

  • Peiping Shen

    (Henan Normal University
    North China University of Water Resources and Electric Power)

Abstract

In this paper, an efficient branch and bound algorithm with a new adjustable branching rule is presented to solve the sum of linear ratios problem (SLRP). In the algorithm, problem (SLRP) is first converted into its equivalent form (ERP) whose objective function involves ( $$p-1$$ p - 1 ) linear ratios via Charnes–Cooper transformation, and (ERP) is equivalently translated to problem (EP) which has a linear objective function by some variables transformation. A convex relaxation problem (CRP) for (EP) is constructed to obtain a lower bound to the optimal value of (EP). In addition, a novel adjustable branching rule is proposed to offer tight lower bounds to the optimal values of (EP) over the corresponding sub-rectangles under some certain conditions. Also, a convex combination method is designed to update the upper bound for the optimal value of (ERP). By continuously refining the initial rectangle and tackling a series of convex relaxation problems, the presented algorithm can find a global optimal solution to (ERP). Moreover, we analyze the complexity result of the proposed algorithm. Finally, the feasibility and effectiveness of the algorithm are verified by preliminary numerical experiments.

Suggested Citation

  • Bingdi Huang & Peiping Shen, 2025. "An efficient global optimization algorithm for the sum of linear ratios problems based on a novel adjustable branching rule," Computational Optimization and Applications, Springer, vol. 91(3), pages 1339-1371, July.
    • Handle: RePEc:spr:coopap:v:91:y:2025:i:3:d:10.1007_s10589-025-00679-8
    DOI: 10.1007/s10589-025-00679-8
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    References listed on IDEAS

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    1. 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.
    2. XiaoLi Huang & YueLin Gao & Bo Zhang & Xia Liu, 2020. "An Effective Computational Algorithm for the Global Solution of a Class of Linear Fractional Programming," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-14, November.
    3. 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).
    4. 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.
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