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Invariant-point theorems and existence of solutions to optimization-related problems

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Abstract

To consider existence of solutions to various optimization-related problems, we first develop some equivalent versions of invariant-point theorems. Next, they are employed to derive sufficient conditions for the solution existence for two general models of variational relation and inclusion problems. We also prove the equivalence of these conditions with the above-mentioned invariant-point theorems. In applications, we include consequences of these results to a wide range of particular cases, from relatively general inclusion problems to classical results as Ekeland’s variational principle, and practical situations like traffic networks and non-cooperative games, to illustrate application possibilities of our general results. Many examples are provided to explain advantages of the obtained results and also to motivate in detail our problem settings.

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Acknowledgments

This work was supported by Vietnam National University Hochiminh City under the grant number B2013-28-01. We are grateful to Dr. Nguyen Ngoc Hai for his valuable remarks and discussions.

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Authors and Affiliations

  1. Department of Mathematics, International University of Hochiminh City, Hochiminh City, Vietnam

    Phan Quoc Khanh

  2. Department of Mathematics, Cao Thang College of Technology, Hochiminh City, Vietnam

    Vo Si Trong Long

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  1. Phan Quoc Khanh
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  2. Vo Si Trong Long
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Correspondence to Vo Si Trong Long.

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Khanh, P.Q., Long, V.S.T. Invariant-point theorems and existence of solutions to optimization-related problems. J Glob Optim 58, 545–564 (2014). https://doi.org/10.1007/s10898-013-0065-y

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  • DOI: https://doi.org/10.1007/s10898-013-0065-y

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