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Theorems of the Alternative and Optimization with Set-Valued Maps

Author

Listed:
  • X. M. Yang

    (Chongqing Normal University)

  • X. Q. Yang

    (Hong Kong Polytechnic University, Hung Hom)

  • G. Y. Chen

    (Chinese Academy of Sciences)

Abstract

In this paper, the concept of generalized cone subconvexlike set-valued mapsis presented and a theorem of alternative for the system of generalizedinequality–equality set-valued maps is established. By applying thetheorem of the alternative and other results, necessary and sufficientoptimality conditions for vector optimization problems with generalizedcone subconvexlike set-valued maps are obtained.

Suggested Citation

  • X. M. Yang & X. Q. Yang & G. Y. Chen, 2000. "Theorems of the Alternative and Optimization with Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 107(3), pages 627-640, December.
    • Handle: RePEc:spr:joptap:v:107:y:2000:i:3:d:10.1023_a:1026407517917
    DOI: 10.1023/A:1026407517917
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    References listed on IDEAS

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    1. Z. Li, 1999. "A Theorem of the Alternative and Its Application to the Optimization of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 100(2), pages 365-375, February.
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    Citations

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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. Z. A. Zhou & X. M. Yang, 2011. "Optimality Conditions of Generalized Subconvexlike Set-Valued Optimization Problems Based on the Quasi-Relative Interior," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 327-340, August.
    5. J. Li & G. Mastroeni, 2016. "Image Convexity of Generalized Systems with Infinite-Dimensional Image and Applications," Journal of Optimization Theory and Applications, Springer, vol. 169(1), pages 91-115, April.
    6. R. Zeng & R. J. Caron, 2006. "Generalized Motzkin Theorems of the Alternative and Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 131(2), pages 281-299, November.
    7. 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.
    8. 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.
    9. 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.

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    1. Z. A. Zhou & X. M. Yang, 2011. "Optimality Conditions of Generalized Subconvexlike Set-Valued Optimization Problems Based on the Quasi-Relative Interior," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 327-340, August.
    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.
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