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A New Global Optimization Algorithm for a Class of Linear Fractional Programming

Author

Listed:
  • X. Liu

    (School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China)

  • Y.L. Gao

    (Ningxia Province Cooperative Innovation Center of Scientific Computing and Intelligent Information Processing, North Minzu University, Yinchuan 750021, China
    Ningxia Province Key Laboratory of Intelligent Information and Data Processing, North Minzu University, Yinchuan 750021, China)

  • B. Zhang

    (School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China)

  • F.P. Tian

    (Ningxia Province Cooperative Innovation Center of Scientific Computing and Intelligent Information Processing, North Minzu University, Yinchuan 750021, China)

Abstract

In this paper, we propose a new global optimization algorithm, which can better solve a class of linear fractional programming problems on a large scale. First, the original problem is equivalent to a nonlinear programming problem: It introduces p auxiliary variables. At the same time, p new nonlinear equality constraints are added to the original problem. By classifying the coefficient symbols of all linear functions in the objective function of the original problem, four sets are obtained, which are I i + , I i − , J i + and J i − . Combined with the multiplication rule of real number operation, the objective function and constraint conditions of the equivalent problem are linearized into a lower bound linear relaxation programming problem. Our lower bound determination method only needs e i T x + f i ≠ 0 , and there is no need to convert molecules to non-negative forms in advance for some special problems. A output-space branch and bound algorithm based on solving the linear programming problem is proposed and the convergence of the algorithm is proved. Finally, in order to illustrate the feasibility and effectiveness of the algorithm, we have done a series of numerical experiments, and show the advantages and disadvantages of our algorithm by the numerical results.

Suggested Citation

  • X. Liu & Y.L. Gao & B. Zhang & F.P. Tian, 2019. "A New Global Optimization Algorithm for a Class of Linear Fractional Programming," Mathematics, MDPI, vol. 7(9), pages 1-21, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:867-:d:268763
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    References listed on IDEAS

    as
    1. 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.
    2. Bartosz Sawik, 2012. "Downside Risk Approach for Multi-Objective Portfolio Optimization," Operations Research Proceedings, in: Diethard Klatte & Hans-Jakob Lüthi & Karl Schmedders (ed.), Operations Research Proceedings 2011, edition 127, pages 191-196, Springer.
    3. Yuelin Gao & Siqiao Jin, 2013. "A Global Optimization Algorithm for Sum of Linear Ratios Problem," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-7, June.
    4. A. Charnes & W. W. Cooper, 1962. "Programming with linear fractional functionals," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 9(3‐4), pages 181-186, September.
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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. 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.
    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. Zhijun Xu & Jing Zhou, 2021. "A Global Optimization Algorithm for Solving Linearly Constrained Quadratic Fractional Problems," Mathematics, MDPI, vol. 9(22), pages 1-12, November.
    5. Hou, Zhisong & Liu, Sanyang, 2023. "A spatial branch-reduction-bound algorithm for solving generalized linear fractional problems globally," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).

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