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Set-Valued Systems with Infinite-Dimensional Image and Applications

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Abstract

In infinite-dimensional spaces, we investigate a set-valued system from the image perspective. By exploiting the quasi-interior and the quasi-relative interior, we give some new equivalent characterizations of (proper, regular) linear separation and establish some new sufficient and necessary conditions for the impossibility of the system under new assumptions, which do not require the set to have nonempty interior. We also present under mild assumptions the equivalence between (proper, regular) linear separation and saddle points of Lagrangian functions for the system. These results are applied to obtain some new saddle point sufficient and necessary optimality conditions of vector optimization problems.

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Acknowledgements

The authors greatly appreciate three anonymous referees for their useful comments and suggestions, which have helped to improve an early version of the paper. The authors also greatly appreciate Dr. G. Mastroeni for his valuable suggestions on the last part of Sect. 5. The research was supported by the National Natural Science Foundation of China, Project 11371015; the Key Project of Chinese Ministry of Education, Project 211163; and Sichuan Youth Science and Technology Foundation, Project 2012JQ0032.

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  1. College of Mathematics and Information, China West Normal University, Nanchong, 637009, Sichuan, China

    J. Li & L. Yang

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  1. J. Li

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  2. L. Yang

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Correspondence toJ. Li.

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Li, J., Yang, L. Set-Valued Systems with Infinite-Dimensional Image and Applications.J Optim Theory Appl179, 868–895 (2018). https://doi.org/10.1007/s10957-016-1041-8

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