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Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps

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  • Z. F. Li

    (University of Inner Mongolia)

Abstract

This paper extends the concept of cone subconvexlikeness of single-valued maps to set-valued maps and presents several equivalent characterizations and an alternative theorem for cone-subconvexlike set-valued maps. The concept and results are then applied to study the Benson proper efficiency for a vector optimization problem with set-valued maps in topological vector spaces. Two scalarization theorems and two Lagrange multiplier theorems are established. After introducing the new concept of proper saddle point for an appropriate set-valued Lagrange map, we use it to characterize the Benson proper efficiency. Lagrange duality theorems are also obtained

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:98:y:1998:i:3:d:10.1023_a:1022676013609
    DOI: 10.1023/A:1022676013609
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    Cited by:

    1. D. S. Kim & G. M. Lee & P. H. Sach, 2004. "Hartley Proper Efficiency in Multifunction Optimization," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 129-145, January.
    2. 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.
    3. P. H. Sach, 2007. "Moreau–Rockafellar Theorems for Nonconvex Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 213-227, May.
    4. Ruchi, Arora & Lalitha, C.S., 2005. "Proximal proper efficiency in set-valued optimization," Omega, Elsevier, vol. 33(5), pages 407-411, October.
    5. Zhenhua Peng & Yihong Xu, 2017. "New Second-Order Tangent Epiderivatives and Applications to Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 128-140, January.
    6. E. Muselli, 2000. "Upper and Lower Semicontinuity for Set-Valued Mappings Involving Constraints," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 527-550, September.
    7. Yi-Hong Xu & Zhen-Hua Peng, 2018. "Second-Order M-Composed Tangent Derivative and Its Applications," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(05), pages 1-20, October.
    8. C. Gutiérrez & L. Huerga & V. Novo & C. Tammer, 2016. "Duality related to approximate proper solutions of vector optimization problems," Journal of Global Optimization, Springer, vol. 64(1), pages 117-139, January.
    9. Y. D. Xu & S. J. Li, 2013. "Optimality Conditions for Generalized Ky Fan Quasi-Inequalities with Applications," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 663-684, June.
    10. A. Taa, 2005. "ɛ-Subdifferentials of Set-valued Maps and ɛ-Weak Pareto Optimality for Multiobjective Optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(2), pages 187-209, November.
    11. Zhiang Zhou & Wenbin Wei & Fei Huang & Kequan Zhao, 2024. "Approximate weak efficiency of the set-valued optimization problem with variable ordering structures," Journal of Combinatorial Optimization, Springer, vol. 48(3), pages 1-13, October.
    12. Ozdemir, Mujgan S. & Gasimov, Rafail N., 2004. "The analytic hierarchy process and multiobjective 0-1 faculty course assignment," European Journal of Operational Research, Elsevier, vol. 157(2), pages 398-408, September.
    13. P. H. Sach, 2003. "Nearly Subconvexlike Set-Valued Maps and Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 335-356, November.
    14. L. Y. Xia & J. H. Qiu, 2008. "Superefficiency in Vector Optimization with Nearly Subconvexlike Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 125-137, January.

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