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Existence of Solutions for Set Optimization Problems with Variable Ordering Structures

Author

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  • Dinh Vinh Hien

    (Ho Chi Minh City University of Industry and Trade)

  • Nguyen Le Hoang Anh

    (University of Science
    Vietnam National University)

Abstract

This article focuses on the existence of solutions to set optimization problems based on variable ordering structures. First, the relationships among different solution types of such problems are disscused. Next, we establish sufficient conditions for the existence of efficient solutions by constructing appropriate level sets and utilizing the semicontinuity properties of objective mappings. Furthermore, we investigate the conditions necessary for the existence of ideal solutions, which require the proper cone-quasiconvexity of the objective mapping. Finally, our analysis extends to weak efficient solutions, applying transfer lower continuity assumptions. This study is novel and enhances the understanding of the results regarding the existence of solutions to set optimization problems in the literature.

Suggested Citation

  • Dinh Vinh Hien & Nguyen Le Hoang Anh, 2025. "Existence of Solutions for Set Optimization Problems with Variable Ordering Structures," Journal of Optimization Theory and Applications, Springer, vol. 207(3), pages 1-20, December.
    • Handle: RePEc:spr:joptap:v:207:y:2025:i:3:d:10.1007_s10957-025-02823-0
    DOI: 10.1007/s10957-025-02823-0
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    References listed on IDEAS

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    1. Lam Quoc Anh & Tran Quoc Duy & Dinh Vinh Hien & Daishi Kuroiwa & Narin Petrot, 2020. "Convergence of Solutions to Set Optimization Problems with the Set Less Order Relation," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 416-432, May.
    2. Stephan Dempe & Maria Pilecka, 2016. "Optimality Conditions for Set-Valued Optimisation Problems Using a Modified Demyanov Difference," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 402-421, November.
    3. 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.
    4. 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.
    5. L. Q. Anh & T. Q. Duy & D. V. Hien, 2020. "Stability of efficient solutions to set optimization problems," Journal of Global Optimization, Springer, vol. 78(3), pages 563-580, November.
    6. Marius Durea & Radu Strugariu & Christiane Tammer, 2015. "On set-valued optimization problems with variable ordering structure," Journal of Global Optimization, Springer, vol. 61(4), pages 745-767, April.
    7. Kuntal Som & V. Vetrivel, 2023. "Global well-posedness of set-valued optimization with application to uncertain problems," Journal of Global Optimization, Springer, vol. 85(2), pages 511-539, February.
    8. Michel H. Geoffroy, 2019. "A topological convergence on power sets well-suited for set optimization," Journal of Global Optimization, Springer, vol. 73(3), pages 567-581, March.
    9. Fatemeh Fakhar & Hamid Reza Hajisharifi & Zeinab Soltani, 2025. "Existence of weak efficient solutions of set-valued optimization problems," Journal of Global Optimization, Springer, vol. 91(1), pages 199-215, January.
    Full references (including those not matched with items on IDEAS)

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