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Connectedness of Solution Sets of Strong Vector Equilibrium Problems with an Application

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Abstract

In this paper, the connectedness of solution set of a strong vector equilibrium problem in a finite dimensional space, is investigated. Firstly, a nonconvex separation theorem is given, that is, a neither open nor closed set and a compact subset in a finite dimensional space can be strictly separated by a sublinear and strongly monotone function. Secondly, in terms of the nonconvex separation theorem, the union relation between the solution set of the strong vector equilibrium problem and the solution sets of a series of nonlinear scalar problems, is established. Under suitable assumptions, the connectedness and the path connectedness of the solution set of the strong vector equilibrium problem are obtained. In particular, we solve partly the question related to the path connectedness of the solution set of the strong vector equilibrium problem. The question is proposed by Han and Huang in the reference (J Optim Theory Appl, 2016. https://doi.org/10.1007/s10957-016-1032-9). Finally, as an application, we apply the main results to derive the connectedness of the solution set of a linear vector optimization problem.

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Acknowledgements

The authors are grateful to Professor Franco Giannessi for their helpful advice and valuable comments, especially for suggestions on adding Section 5 and Section 6, which helped to improve the paper. They would also like to express their gratitude to two anonymous referees for their valuable comments and suggestions. This research was supported by the National Natural Science Foundation of China (Grant Numbers: 11426055, 11601054), the Basic and Advanced Research Project of CQ CSTC (Grant Numbers: cstc2014jcyjA00044, cstc2016jcyjA0466) and the Science and Technology Research Project of Chongqing Municipal Education Commission (Grant Numbers: KJ1500419, KJ1600403) and the Chinese Scholarship Council (Grant Number: CSC201607845012).

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  1. College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Yangdong Xu & Pingping Zhang

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  1. Yangdong Xu
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  2. Pingping Zhang
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Correspondence to Yangdong Xu.

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Xu, Y., Zhang, P. Connectedness of Solution Sets of Strong Vector Equilibrium Problems with an Application. J Optim Theory Appl 178, 131–152 (2018). https://doi.org/10.1007/s10957-018-1244-2

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