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Nonlinear Separation Approach to Inverse Variational Inequalities in Real Linear Spaces

Author

Listed:
  • Elisabeth Köbis

    (Martin Luther University Halle-Wittenberg)

  • Markus A. Köbis

    (Free University Berlin)

  • Xiaolong Qin

    (College of Mathematics and Computer Science, Zhejiang Normal University)

Abstract

In this paper, we employ the image space analysis to study constrained inverse variational inequalities by means of a nonlinear separation approach. We introduce such a nonlinear functional, which is based on the known Gerstewitz functional, and show its property as a weak separation function and a regular weak separation function under different parameter sets. In contrast to known results, we do not assume any topology on the considered spaces. Then, an alternative theorem is established, which leads directly to a sufficient and necessary optimality condition of the constrained inverse variational inequality. Finally, a gap function and an error bound are obtained for the constrained inverse variational inequality.

Suggested Citation

  • Elisabeth Köbis & Markus A. Köbis & Xiaolong Qin, 2019. "Nonlinear Separation Approach to Inverse Variational Inequalities in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 105-121, October.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:1:d:10.1007_s10957-019-01543-6
    DOI: 10.1007/s10957-019-01543-6
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    References listed on IDEAS

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    1. 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.
    2. Jiawei Chen & Elisabeth Köbis & Markus Köbis & Jen-Chih Yao, 2018. "Image Space Analysis for Constrained Inverse Vector Variational Inequalities via Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 816-834, June.
    3. S. J. Li & Y. D. Xu & S. K. Zhu, 2012. "Nonlinear Separation Approach to Constrained Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 842-856, September.
    4. He, Bingsheng & He, Xiao-Zheng & Liu, Henry X., 2010. "Solving a class of constrained 'black-box' inverse variational inequalities," European Journal of Operational Research, Elsevier, vol. 204(3), pages 391-401, August.
    5. Klamroth, Kathrin & Köbis, Elisabeth & Schöbel, Anita & Tammer, Christiane, 2017. "A unified approach to uncertain optimization," European Journal of Operational Research, Elsevier, vol. 260(2), pages 403-420.
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