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Journal of Optimization Theory and Applications

, Volume 180, Issue 1, pp 187–206 | Cite as

Optimality Conditions for Vector Equilibrium Problems with Applications

  • Alfredo IusemEmail author
  • Felipe Lara
Article
  • 179 Downloads

Abstract

We use asymptotic analysis for studying noncoercive pseudomonotone equilibrium problems and vector equilibrium problems. We introduce suitable notions of asymptotic functions, which provide sufficient conditions for the set of solutions of these problems to be nonempty and compact under quasiconvexity of the objective function. We characterize the efficient and weakly efficient solution set for the nonconvex vector equilibrium problem via scalarization. A sufficient condition for the closedness of the image of a nonempty, closed and convex set via a quasiconvex vector-valued function is given. Finally, applications to the quadratic fractional programming problem are also presented.

Keywords

Asymptotic analysis Generalized convexity Pseudomonotone operators Equilibrium problems Vector optimization 

Mathematics Subject Classification

90C20 90C26 90C32 

Notes

Acknowledgements

The authors want to express their gratitude to both referees for their criticism and suggestions that helped to improve this paper. The research for the second author was partially supported by Conicyt–Chile under project Fondecyt Postdoctorado 3160205.

References

  1. 1.
    Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972)Google Scholar
  2. 2.
    Brézis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan’s minimax principle. Bolletino della Unione Matematica Italiana 6(4), 293–300 (1972)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Oettli, W.: A remark on vector-valued equilibria and generalized monotonicity. Acta Math. Vietnam. 22, 213–221 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Flores-Bazán, F.: Existence theory for finite-dimensional pseudomonotone equilibrium problems. Acta Appl. Math. 77, 249–297 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ait Mansour, M., Chbani, Z., Riahi, H.: Recession bifunction and solvability of noncoercive equilibrium problems. Commun. Appl. Anal. 7, 369–377 (2003)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Iusem, A., Kassay, G., Sosa, W.: On certain conditions for the existence of solutions of equilibrium problems. Math. Program. 116, 259–273 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Aussel, D., Cotrina, J., Iusem, A.: An existence result for quasi-equilibrium problems. J. Convex Anal. 24, 55–66 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  10. 10.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  11. 11.
    Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2003)zbMATHGoogle Scholar
  12. 12.
    Amara, C.: Directions de majoration d’une fonction quasiconvexe et applications. Serdica Math. J. 24, 289–306 (1998)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Penot, J.P.: What is quasiconvex analysis? Optimization 47, 35–110 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivates. J. Glob. Optim. 63, 99–123 (2015)CrossRefzbMATHGoogle Scholar
  15. 15.
    Flores-Bazán, F., Hadjisavvas, N., Lara, F., Montenegro, I.: First- and second-order asymptotic analysis with applications in quasiconvex optimization. J. Optim. Theory Appl. 170, 372–393 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lara, F., López, R.: Formulas for asymptotic functions via conjugates, directional derivatives and subdifferentials. J. Optim. Theory Appl. 173, 793–811 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Iusem, A., Lara, F.: The \(q\)-asympotic function in \(c\)-convex analysis. Optimization (2018).  https://doi.org/10.1080/02331934.2018.1456540
  18. 18.
    Lara, F.: Generalized asymptotic functions in nonconvex multiobjective optimization problems. Optimization 66, 1259–1272 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Attouch, H., Chbani, Z., Moudafi, A.: Recession operators and solvability of variational problems in reflexive Banach spaces. In: Bauchitté, G., et al. (eds.) Calculus of Variations, Homogenization and Continuum Mechanics, pp. 51–67. World Scientific, Singapore (1994)Google Scholar
  20. 20.
    Deng, S.: Coercivity properties and well-posedness in vector optimization. RAIRO Oper. Res. 37, 195–208 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Deng, S.: Boundedness and nonemptiness of the efficient solution sets in multiobjective optimization. J. Optim. Theory Appl. 144, 29–42 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hadjisavvas, N., Schaible, S.: Quasimonotonicity and pseudomonotonicity in variational inequalities and equilibrium problems. In: Crouzeix, J.P., et al. (eds.) Generalized Convexity, Generalized Monotonicity: Recent Results, pp. 257–275. Kluwer, Dordrech (1998)CrossRefGoogle Scholar
  23. 23.
    Iusem, A., Lara, F.: Second order asympotic functions and applications to quadratic programming. J. Convex Anal. 25, 271–291 (2018)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Cambini, A., Martein, L.: Generalized Convexity and Optimization. Springer, Berlin (2009)zbMATHGoogle Scholar
  25. 25.
    Hadjisavvas, N., Komlosi, S., Schaible, S.: Handbook of Generalized Convexity and Generalized Monotonicity. Springer, Boston (2005)CrossRefzbMATHGoogle Scholar
  26. 26.
    Giannessi, F.: Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    Ansari, Q.H., Yao, J.C.: Recent Developments in Vector Optimization. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  28. 28.
    Jeyakumar, V., Oettli, W., Natividad, M.: A solvability theorem for a class of quasiconvex mappings with applications to optimization. J. Math. Anal. Appl. 179, 537–546 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kuroiwa, D.: Convexity for set-valued maps. Appl. Math. Lett. 9, 97–101 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sawaragi, Y., Nakayama, H., Tanino, Y.: Theory of Multiobjective Optimization. Academic Press, New York (1985)zbMATHGoogle Scholar
  31. 31.
    Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, pp. 495–608. Kluwer Academic Publishers, Dordrecht (1995)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto Nacional de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil
  2. 2.Departamento de MatemáticasUniversidad de TarapacáAricaChile

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