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Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications

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  • Chuang-Liang Zhang

    (Sichuan University
    Jiaying University)

  • Nan-jing Huang

    (Sichuan University)

Abstract

In this paper, we give some properties concerned with weak minimal solutions of nonconvex set optimization problems. We also give some properties of the nonconvex separation functional and apply them to characterize weak minimal solutions of nonconvex set optimization problems. Moreover, we derive some new existence results for weak minimal solutions of nonconvex set optimization problems whose image spaces have no topology. Finally, we establish a set-valued version of Ekeland’s variational principle via set relations and present a weak minimization for a nonconvex set optimization problem. As applications, we obtain the existence of weak minimal solutions for nonconvex vector optimization problems.

Suggested Citation

  • Chuang-Liang Zhang & Nan-jing Huang, 2021. "Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 894-914, September.
    • Handle: RePEc:spr:joptap:v:190:y:2021:i:3:d:10.1007_s10957-021-01913-z
    DOI: 10.1007/s10957-021-01913-z
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    References listed on IDEAS

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    1. Petra Weidner, 2017. "Gerstewitz Functionals on Linear Spaces and Functionals with Uniform Sublevel Sets," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 812-827, June.
    2. Yu Han & Nan-jing Huang, 2018. "Continuity and Convexity of a Nonlinear Scalarizing Function in Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 679-695, June.
    3. M. Chinaie & F. Fakhar & M. Fakhar & H. R. Hajisharifi, 2019. "Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem," Journal of Global Optimization, Springer, vol. 75(1), pages 131-141, September.
    4. Adan, M. & Novo, V., 2003. "Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness," European Journal of Operational Research, Elsevier, vol. 149(3), pages 641-653, September.
    5. 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.
    6. L. P. Hai & L. Huerga & P. Q. Khanh & V. Novo, 2019. "Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems," Journal of Global Optimization, Springer, vol. 74(2), pages 361-382, June.
    7. Khushboo & C. S. Lalitha, 2019. "A unified minimal solution in set optimization," Journal of Global Optimization, Springer, vol. 74(1), pages 195-211, May.
    8. Truong Quang Bao & Christiane Tammer, 2019. "Scalarization Functionals with Uniform Level Sets in Set Optimization," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 310-335, July.
    9. C. Gutiérrez & L. Huerga & B. Jiménez & V. Novo, 2018. "Approximate solutions of vector optimization problems via improvement sets in real linear spaces," Journal of Global Optimization, Springer, vol. 70(4), pages 875-901, April.
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