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Outer space branch and bound algorithm for solving linear multiplicative programming problems

Author

Listed:
  • Peiping Shen

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

  • Kaimin Wang

    (Henan Normal University)

  • Ting Lu

    (Henan Normal University)

Abstract

In this paper, we consider a linear multiplicative programming problem (LMP) that is known to be NP-hard even with one product term. We first introduce the auxiliary variables to obtain an equivalent problem of problem LMP. An outer space branch and bound algorithm is then designed, which integrates some basic operations such as the linear relaxation technique, branching rule and region reduction technique. The global convergence of the proposed algorithm is established by means of the subsequent solutions of a series of linear programming problems, and its computational complexity is estimated on the basis of the branching rule. Also, we discuss the relationship between the proposed linear relaxation and existing relaxations for LMP. Finally, preliminary numerical results demonstrate the proposed algorithm can efficiently find the globally optimal solutions for test instances.

Suggested Citation

  • Peiping Shen & Kaimin Wang & Ting Lu, 2020. "Outer space branch and bound algorithm for solving linear multiplicative programming problems," Journal of Global Optimization, Springer, vol. 78(3), pages 453-482, November.
    • Handle: RePEc:spr:jglopt:v:78:y:2020:i:3:d:10.1007_s10898-020-00919-7
    DOI: 10.1007/s10898-020-00919-7
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    Cited by:

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    2. Yunchol Jong & Yongjin Kim & Hyonchol Kim, 2024. "A method based on parametric convex programming for solving convex multiplicative programming problem," Journal of Global Optimization, Springer, vol. 90(3), pages 573-592, November.
    3. Bo Zhang & Hongyu Wang & Yuelin Gao, 2024. "Output-Space Outer Approximation Branch-and-Bound Algorithm for a Class of Linear Multiplicative Programs," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 997-1026, September.
    4. 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.
    5. R. Cambini & R. Riccardi & D. Scopelliti, 2023. "Solving linear multiplicative programs via branch-and-bound: a computational experience," Computational Management Science, Springer, vol. 20(1), pages 1-32, December.
    6. Gao, YueLin & Zhang, Bo, 2023. "Output-space branch-and-bound reduction algorithm for generalized linear fractional-multiplicative programming problem," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    7. 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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