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Linear and Nonlinear Scalarization Methods for Vector Optimization Problems with Variable Ordering Structures

Author

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  • Jian-Wen Peng

    (Chongqing Normal University)

  • Wen-Bin Wei

    (Chongqing Normal University)

  • Refail Kasimbeyli

    (Eskisehir Technical University
    Azerbaijan State University of Economics)

Abstract

This paper investigates linear and nonlinear scalarization methods for vector optimization problems with variable ordering structures (VOS). Firstly, we introduce the concepts of $$\varepsilon $$ ε -efficient elements and weakly $$\varepsilon $$ ε -efficient elements of a set with VOSs given by coradiant sets. Secondly we derive characterization theorems for weakly $$\varepsilon $$ ε -efficient solutions in the sense of linear scalarization. Then, we establish characterization theorems for weakly $$\varepsilon $$ ε -efficient solutions in the sense of nonlinear scalarization via the Hirriart-Urruty nonlinear functions and the functions defined via the Kasimbeyli’s augmented dual cones. Finally, we establish nonlinear scalarization theorems for the weakly $$\varepsilon $$ ε -efficient elements of a set via the augmented dual cones approach. The results of this paper generalize the corresponding results in the literature.

Suggested Citation

  • Jian-Wen Peng & Wen-Bin Wei & Refail Kasimbeyli, 2025. "Linear and Nonlinear Scalarization Methods for Vector Optimization Problems with Variable Ordering Structures," Journal of Optimization Theory and Applications, Springer, vol. 206(1), pages 1-21, July.
  • Handle: RePEc:spr:joptap:v:206:y:2025:i:1:d:10.1007_s10957-025-02662-z
    DOI: 10.1007/s10957-025-02662-z
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    References listed on IDEAS

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    1. Truong Q. Bao & Boris S. Mordukhovich & Antoine Soubeyran & Christiane Tammer, 2022. "Vector Optimization with Domination Structures: Variational Principles and Applications," Post-Print hal-03528619, HAL.
    2. Refail Kasimbeyli, 2013. "A conic scalarization method in multi-objective optimization," Journal of Global Optimization, Springer, vol. 56(2), pages 279-297, June.
    3. Z. F. Li, 1998. "Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 623-649, September.
    4. Abbas Sayadi-bander & Latif Pourkarimi & Refail Kasimbeyli & Hadi Basirzadeh, 2017. "Coradiant sets and $$\varepsilon $$ ε -efficiency in multiobjective optimization," Journal of Global Optimization, Springer, vol. 68(3), pages 587-600, July.
    5. C. Gutiérrez & B. Jiménez & V. Novo, 2006. "On Approximate Efficiency in Multiobjective Programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 165-185, August.
    6. Refail Kasimbeyli & Masoud Karimi, 2021. "Duality in nonconvex vector optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 139-160, May.
    7. Gabriele Eichfelder & Refail Kasimbeyli, 2014. "Properly optimal elements in vector optimization with variable ordering structures," Journal of Global Optimization, Springer, vol. 60(4), pages 689-712, December.
    8. Shokouh Shahbeyk & Majid Soleimani-damaneh & Refail Kasimbeyli, 2018. "Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure," Journal of Global Optimization, Springer, vol. 71(2), pages 383-405, June.
    9. Z. A. Zhou & J. W. Peng, 2012. "Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 830-841, September.
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    11. Gutiérrez, C. & Jiménez, B. & Novo, V., 2012. "Improvement sets and vector optimization," European Journal of Operational Research, Elsevier, vol. 223(2), pages 304-311.
    12. Gabriele Eichfelder & Maria Pilecka, 2016. "Set Approach for Set Optimization with Variable Ordering Structures Part I: Set Relations and Relationship to Vector Approach," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 931-946, December.
    13. Behnam Soleimani, 2014. "Characterization of Approximate Solutions of Vector Optimization Problems with a Variable Order Structure," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 605-632, August.
    14. Gabriele Eichfelder & Maria Pilecka, 2016. "Set Approach for Set Optimization with Variable Ordering Structures Part II: Scalarization Approaches," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 947-963, December.
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