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New Generalized Convexity Notion for Set-Valued Maps and Application to Vector Optimization

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  • P. H. Sach

    (Professor, Institute of Mathematics)

Abstract

In this paper, we introduce a new generalized convexity notion for set-valued maps, called ic-cone-convexlikeness, and use it as the main tool to derive an alternative theorem and necessary conditions for efficient, weakly efficient, and Benson properly efficient solutions of the problem of minimizing a set-valued map subject to set-valued constraints. Our results are valid for a class of optimization problems broader than that of the problems considered in Refs. 1--6 and generalize the corresponding results of these references.

Suggested Citation

  • P. H. Sach, 2005. "New Generalized Convexity Notion for Set-Valued Maps and Application to Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 125(1), pages 157-179, April.
    • Handle: RePEc:spr:joptap:v:125:y:2005:i:1:d:10.1007_s10957-004-1716-4
    DOI: 10.1007/s10957-004-1716-4
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    Cited by:

    1. 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.
    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. 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.
    5. P. Q. Khanh & N. D. Tuan, 2008. "Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 243-261, November.

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