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Optimization Letters

, Volume 12, Issue 3, pp 615–623 | Cite as

A note on existence of weak efficient solutions for vector equilibrium problems

  • C. Gutiérrez
  • V. Novo
  • J. L. Ródenas-Pedregosa
Original Paper
  • 171 Downloads

Abstract

This work concerns with vector equilibrium problems where the image space of the bifunction is not endowed with any topology. To be precise, a kind of “semi-algebraic” upper semicontinuity notion is introduced and, by means of a recent algebraic version of the so-called Gerstewitz’s functional, a new existence result of weak efficient solutions is obtained that significantly improves some previous ones stated in the topological setting since it requires weaker assumptions.

Keywords

Vector equilibrium problem Algebraic interior Gerstewitz’s functional Triangle inequality property Semicontinuity 

Notes

Acknowledgements

The authors are grateful to the anonymous referee for his/her useful suggestions and remarks. This work was partially supported by Ministerio de Economía y Competitividad (Spain) under Project MTM2015-68103-P (MINECO/FEDER) and by ETSI Industriales, Universidad Nacional de Educación a Distancia (Spain) under Grant 2017-Mat11. Third author was also supported by Spanish FPI Fellowship Programme (BES-2013-066316).

References

  1. 1.
    Ansari, Q.H.: Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory. J. Math. Anal. Appl. 334, 561–575 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ansari, Q.H., Oettli, W., Schläger, D.: A generalization of vectorial equilibria. Math. Methods Oper. Res. 46, 147–152 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Araya, Y., Kimura, K., Tanaka, T.: Existence of vector equilibria via Ekeland’s variational principle. Taiwan. J. Math. 12, 1991–2000 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bianchi, M., Kassay, G., Pini, R.: Ekeland’s principle for vector equilibrium problems. Nonlinear Anal. 66, 1454–1464 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Nonconvex Optim. Appl., vol. 38. Kluwer, Dordrecht (2000)Google Scholar
  8. 8.
    Göpfert, A., Rihai, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)zbMATHGoogle Scholar
  9. 9.
    Gutiérrez, C., Kassay, G., Novo, V., Ródenas-Pedregosa, J.L.: Ekeland variational principles in vector equilibrium problems. SIAM J. Optim. 27, 2405–2425 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gutiérrez, C., Novo, V., Ródenas-Pedregosa, J.L., Tanaka, T.: Nonconvex separation functional in linear spaces with applications to vector equilibria. SIAM J. Optim. 26, 2677–2695 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2011)zbMATHGoogle Scholar
  12. 12.
    Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization. An Introduction with Applications. Springer, Berlin (2015)zbMATHGoogle Scholar
  13. 13.
    Oettli, W.: A remark on vector-valued equilibria and generalized monotonicity. Acta Math. Vietnam. 22, 213–221 (1997)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Penot, J.P., Thera, M.: Semi-continuous mappings in general topology. Arch. Math. 38, 158–166 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Qiu, J.H., He, F.: A general vectorial Ekeland’s variational principle with a P-distance. Acta Math. Sin. (Engl. Ser.) 29, 1655–1678 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tammer, C.: A generalization of Ekeland’s variational principle. Optimization 25, 129–141 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zeng, J., Li, S.J.: An Ekeland’s variational principle for set-valued mappings with applications. J. Comput. Appl. Math. 230, 477–484 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • C. Gutiérrez
    • 1
  • V. Novo
    • 2
  • J. L. Ródenas-Pedregosa
    • 2
  1. 1.IMUVA (Institute of Mathematics of University of Valladolid)ValladolidSpain
  2. 2.Departamento de Matemática Aplicada, E.T.S.I. IndustrialesUniversidad Nacional de Educación a DistanciaMadridSpain

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