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A new scheme for approximating the weakly efficient solution set of vector rational optimization problems

Author

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  • Feng Guo

    (Dalian University of Technology)

  • Liguo Jiao

    (Northeast Normal University
    Shanghai Zhangjiang Academy of Mathematics)

Abstract

In this paper, we provide a new scheme for approximating the weakly efficient solution set for a class of vector optimization problems with rational objectives over a feasible set defined by finitely many polynomial inequalities. More precisely, we present a procedure to obtain a sequence of explicit approximations of the weakly efficient solution set of the problem in question. Each approximation is the intersection of the sublevel set of a single polynomial and the feasible set. To this end, we make use of the achievement function associated with the considered problem and construct polynomial approximations of it over the feasible set from above. Remarkably, the construction can be converted to semidefinite programming problems. Several nontrivial examples are designed to illustrate the proposed new scheme.

Suggested Citation

  • Feng Guo & Liguo Jiao, 2023. "A new scheme for approximating the weakly efficient solution set of vector rational optimization problems," Journal of Global Optimization, Springer, vol. 86(4), pages 905-930, August.
    • Handle: RePEc:spr:jglopt:v:86:y:2023:i:4:d:10.1007_s10898-023-01287-8
    DOI: 10.1007/s10898-023-01287-8
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    1. Jonathan M. Borwein, 1983. "On the Existence of Pareto Efficient Points," Mathematics of Operations Research, INFORMS, vol. 8(1), pages 64-73, February.
    2. Metev, Boyan & Gueorguieva, Dessislava, 2000. "A simple method for obtaining weakly efficient points in multiobjective linear fractional programming problems," European Journal of Operational Research, Elsevier, vol. 126(2), pages 386-390, October.
    3. Hiroki Tanabe & Ellen H. Fukuda & Nobuo Yamashita, 2019. "Proximal gradient methods for multiobjective optimization and their applications," Computational Optimization and Applications, Springer, vol. 72(2), pages 339-361, March.
    4. Abbas Amini Fasakhodi & Seyed Nouri & Manouchehr Amini, 2010. "Water Resources Sustainability and Optimal Cropping Pattern in Farming Systems; A Multi-Objective Fractional Goal Programming Approach," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 24(15), pages 4639-4657, December.
    5. Gorissen, B.L. & den Hertog, D., 2012. "Approximating the Pareto Set of Multiobjective Linear Programs via Robust Optimization," Other publications TiSEM 666c5307-4a4e-4be4-a0d0-b, Tilburg University, School of Economics and Management.
    6. C. Veeramani & S. A. Edalatpanah & S. Sharanya & Dragan PamuÄ ar, 2021. "Solving the Multiobjective Fractional Transportation Problem through the Neutrosophic Goal Programming Approach," Discrete Dynamics in Nature and Society, Hindawi, vol. 2021, pages 1-17, August.
    7. N. T. T. Huong & J.-C. Yao & N. D. Yen, 2020. "Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets," Journal of Global Optimization, Springer, vol. 78(3), pages 545-562, November.
    8. Jonathan S. H. Kornbluth & Ralph E. Steuer, 1981. "Multiple Objective Linear Fractional Programming," Management Science, INFORMS, vol. 27(9), pages 1024-1039, September.
    9. Gutjahr, Walter J. & Nolz, Pamela C., 2016. "Multicriteria optimization in humanitarian aid," European Journal of Operational Research, Elsevier, vol. 252(2), pages 351-366.
    10. Jae Hyoung Lee & Liguo Jiao, 2018. "Solving Fractional Multicriteria Optimization Problems with Sum of Squares Convex Polynomial Data," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 428-455, February.
    11. Boaz Golany & Steven Hackman & Ury Passy, 2006. "An efficiency measurement framework for multi-stage production systems," Annals of Operations Research, Springer, vol. 145(1), pages 51-68, July.
    12. Xiaopeng Zhao & Markus A. Köbis & Yonghong Yao & Jen-Chih Yao, 2021. "A Projected Subgradient Method for Nondifferentiable Quasiconvex Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 82-107, July.
    13. Mojtaba Borza & Azmin Sham Rambely, 2021. "A New Method to Solve Multi-Objective Linear Fractional Problems," Fuzzy Information and Engineering, Taylor & Francis Journals, vol. 13(3), pages 323-334, July.
    14. Gorissen, B.L. & den Hertog, D., 2012. "Approximating the Pareto Set of Multiobjective Linear Programs via Robust Optimization," Discussion Paper 2012-031, Tilburg University, Center for Economic Research.
    15. Xiaopeng Zhao & Jen-Chih Yao, 2022. "Linear convergence of a nonmonotone projected gradient method for multiobjective optimization," Journal of Global Optimization, Springer, vol. 82(3), pages 577-594, March.
    16. Thai Doan Chuong, 2022. "Second-order cone programming relaxations for a class of multiobjective convex polynomial problems," Annals of Operations Research, Springer, vol. 311(2), pages 1017-1033, April.
    17. Jae Hyoung Lee & Nithirat Sisarat & Liguo Jiao, 2021. "Multi-objective convex polynomial optimization and semidefinite programming relaxations," Journal of Global Optimization, Springer, vol. 80(1), pages 117-138, May.
    18. Werner Dinkelbach, 1967. "On Nonlinear Fractional Programming," Management Science, INFORMS, vol. 13(7), pages 492-498, March.
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