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On topological existence theorems and applications to optimization-related problems

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  • Phan Khanh
  • Lai Lin
  • Vo Long

Abstract

In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures. Then, we study the solution existence of a number of optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Phan Khanh & Lai Lin & Vo Long, 2014. "On topological existence theorems and applications to optimization-related problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 79(3), pages 253-272, June.
    • Handle: RePEc:spr:mathme:v:79:y:2014:i:3:p:253-272
    DOI: 10.1007/s00186-014-0462-0
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    References listed on IDEAS

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    1. Yannelis, Nicholas C. & Prabhakar, N. D., 1983. "Existence of maximal elements and equilibria in linear topological spaces," Journal of Mathematical Economics, Elsevier, vol. 12(3), pages 233-245, December.
    2. Q. H. Ansari & Y. C. Lin & J. C. Yao, 2000. "General KKM Theorem with Applications to Minimax and Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 104(1), pages 17-57, January.
    3. 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.
    4. 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.
    5. 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.
    6. D. T. Luc, 2008. "An Abstract Problem in Variational Analysis," Journal of Optimization Theory and Applications, Springer, vol. 138(1), pages 65-76, July.
    7. Lai-Jiu Lin, 2012. "Variational relation problems and equivalent forms of generalized Fan-Browder fixed point theorem with applications to Stampacchia equilibrium problems," Journal of Global Optimization, Springer, vol. 53(2), pages 215-229, June.
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