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Vector Variational-Like Inequalities with Constraints: Separation and Alternative

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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.

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Acknowledgments

The authors would like to thank the associated editor and the two anonymous referees for their valuable comments and suggestions, which have helped to improve the paper. This research was partially supported by the Natural Science Foundation of China (Grants: 11171362,11401487,71471140), the Fundamental Research Funds for the Central Universities (Grants: SWU113037, XDJK2014C073), the China Postdoctoral Science Foundation and the Grant MOST 103-2923-E-037-001-MY3.

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Authors and Affiliations

  1. College of Computer Science, Chongqing University, Chongqing, 400044, China

    Jiawei Chen

  2. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Jiawei Chen

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Shengjie Li

  4. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Zhongping Wan

  5. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Jen-Chih Yao

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  1. Jiawei Chen
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  2. Shengjie Li
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  3. Zhongping Wan
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  4. Jen-Chih Yao
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Correspondence to Shengjie Li.

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Communicated by Guang-ya Chen.

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Chen, J., Li, S., Wan, Z. et al. Vector Variational-Like Inequalities with Constraints: Separation and Alternative. J Optim Theory Appl 166, 460–479 (2015). https://doi.org/10.1007/s10957-015-0736-6

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  • DOI: https://doi.org/10.1007/s10957-015-0736-6

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