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

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  • Phan Khanh
  • Vo Long

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. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Phan Khanh & Vo Long, 2014. "Invariant-point theorems and existence of solutions to optimization-related problems," Journal of Global Optimization, Springer, vol. 58(3), pages 545-564, March.
    • Handle: RePEc:spr:jglopt:v:58:y:2014:i:3:p:545-564
    DOI: 10.1007/s10898-013-0065-y
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    References listed on IDEAS

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    1. L. Q. Anh & P. Q. Khanh, 2007. "On the Stability of the Solution Sets of General Multivalued Vector Quasiequilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 135(2), pages 271-284, November.
    2. K. R. Kazmi & S. A. Khan, 2009. "Existence of Solutions to a Generalized System," Journal of Optimization Theory and Applications, Springer, vol. 142(2), pages 355-361, August.
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    5. P. Q. Khanh & N. H. Quan, 2010. "Existence Results for General Inclusions Using Generalized KKM Theorems with Applications to Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 640-653, September.
    6. Nguyen Hai & Phan Khanh & Nguyen Quan, 2009. "On the existence of solutions to quasivariational inclusion problems," Computational Optimization and Applications, Springer, vol. 45(4), pages 565-581, December.
    7. M. Bianchi & I. Konnov & R. Pini, 2010. "Lexicographic and sequential equilibrium problems," Journal of Global Optimization, Springer, vol. 46(4), pages 551-560, April.
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    10. P. Khanh & D. Quy, 2011. "On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings," Journal of Global Optimization, Springer, vol. 49(3), pages 381-396, March.
    11. M. Balaj & L. J. Lin, 2011. "Generalized Variational Relation Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 1-13, January.
    12. P. Q. Khanh & L. M. Luu, 2004. "On the Existence of Solutions to Vector Quasivariational Inequalities and Quasicomplementarity Problems with Applications Break to Traffic Network Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 123(3), pages 533-548, December.
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    Cited by:

    1. Vo Si Trong Long, 2022. "An Invariant-Point Theorem in Banach Space with Applications to Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 440-464, August.

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