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Scalarization of $$\epsilon $$ ϵ -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces

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  • Zhi-Ang Zhou

    (College of Mathematics and Statistics, Chongqing University of Technology)

  • Xin-Min Yang

    (School of Mathematics, Chongqing Normal University)

Abstract

In this paper, we investigate the scalarization of $$\epsilon $$ ϵ -super efficient solutions of set-valued optimization problems in real ordered linear spaces. First, in real ordered linear spaces, under the assumption of generalized cone subconvexlikeness of set-valued maps, a dual decomposition theorem is established in the sense of $$\epsilon $$ ϵ -super efficiency. Second, as an application of the dual decomposition theorem, a linear scalarization theorem is given. Finally, without any convexity assumption, a nonlinear scalarization theorem characterized by the seminorm is obtained.

Suggested Citation

  • Zhi-Ang Zhou & Xin-Min Yang, 2014. "Scalarization of $$\epsilon $$ ϵ -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 680-693, August.
    • Handle: RePEc:spr:joptap:v:162:y:2014:i:2:d:10.1007_s10957-014-0565-z
    DOI: 10.1007/s10957-014-0565-z
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    References listed on IDEAS

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    1. 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.
    2. X. Y. Zheng, 1997. "Proper Efficiency in Locally Convex Topological Vector Spaces," Journal of Optimization Theory and Applications, Springer, vol. 94(2), pages 469-486, August.
    3. Taiyong Li & Yihong Xu & Chuanxi Zhu, 2007. "∊-Strictly Efficient Solutions Of Vector Optimization Problems With Set-Valued Maps," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 24(06), pages 841-854.
    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. M. Adán & V. Novo, 2004. "Proper Efficiency in Vector Optimization on Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 515-540, June.
    6. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
    7. W. D. Rong & Y. N. Wu, 2000. "∈-Weak Minimal Solutions of Vector Optimization Problems with Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 569-579, September.
    8. 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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    Cited by:

    1. Elisabeth Köbis & Markus A. Köbis & Xiaolong Qin, 2020. "An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces," Mathematics, MDPI, vol. 8(1), pages 1-17, January.

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