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Vector Variational-Like Inequalities with Constraints: Separation and Alternative

Author

Listed:
  • Jiawei Chen

    (Chongqing University
    Southwest University)

  • Shengjie Li

    (Chongqing University)

  • Zhongping Wan

    (Wuhan University)

  • Jen-Chih Yao

    (China Medical University)

Abstract

Based on the oriented distance function, a linear weak separation function and three nonlinear regular weak separation functions are introduced in reflexive Banach spaces. Particularly, a nonlinear regular weak separation function does not involve any parameters. Moreover, theorems of the weak alternative for vector variational-like inequalities with constraints are derived by the separation functions without any convexity. Saddle-point conditions, which show the equivalence between the existence of a saddle point and a (linear) nonlinear separation of two suitable subsets of the image space, are established for the linear and nonlinear regular weak separation functions, respectively. Necessary and sufficient optimality conditions for vector variational-like inequalities with constraints are also obtained via the saddle-point conditions. Finally, two gap functions for vector variational-like inequalities with constraints and their continuity are derived by using the image space analysis.

Suggested Citation

  • Jiawei Chen & Shengjie Li & Zhongping Wan & Jen-Chih Yao, 2015. "Vector Variational-Like Inequalities with Constraints: Separation and Alternative," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 460-479, August.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:2:d:10.1007_s10957-015-0736-6
    DOI: 10.1007/s10957-015-0736-6
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    References listed on IDEAS

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    Cited by:

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    2. Jiawei Chen & La Huang & Shengjie Li, 2018. "Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 794-823, September.
    3. 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.
    4. Qamrul Hasan Ansari & Elisabeth Köbis & Pradeep Kumar Sharma, 2019. "Characterizations of Multiobjective Robustness via Oriented Distance Function and Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 817-839, June.

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