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Gap Functions for Vector Equilibrium Problems via Conjugate Duality

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Optimization and Optimal Control

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 39))

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Summary

This chapter deals with the so-called perturbation approach in the conjugate duality for vector optimization on the basis of weak orderings. As applications, we investigate some new set-valued gap functions for vector equilibrium problems.

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References

  1. Altangerel, L., Boţ, R.I., Wanka, G.: On gap functions for equilibrium problems via Fenchel duality. Paci. J. Optim. 2(3), 667–678 (2006)

    MATH  Google Scholar 

  2. Altangerel, L., Boţ, R.I., Wanka, G.: Conjugate duality in vector optimization and some applications to the vector variational inequality. J. Math. Anal. Appl. 329(2), 1010–1035 (2007a)

    Article  MathSciNet  MATH  Google Scholar 

  3. Altangerel, L., Boţ, R.I., Wanka, G. Variational principles for vector equilibrium problems related to conjugate duality. J. Nonlinear. Convex Anal. 8(2) 179–196 (2007b)

    MathSciNet  MATH  Google Scholar 

  4. Ansari, Q.H., Konnov, I.V., Yao, J.C. Existence of a solution and variational principles for vector equilibrium problems. J. Optim. Theory Appl. 110(3), 481–492 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ansari, Q.H., Konnov, I.V., Yao, J.C. Characterizations of solutions for vector equilibrium problems. J. Optimi. Theory Appl. 113(3), 435–447 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blum, E., Oettli, W. Variational principles for equilibrium problems. In: J. Guddat, (Ed.) et al., Parametric Optimization and Related Topics. III. Proceedings of the 3rd conference held in Güstrow, Germany, Frankfurt am Main: Peter Lang Verlag. Approximation Optimization. 3, 79–88 (1993)

    Google Scholar 

  7. Chen, G.Y., Goh, C.J., Yang, X.Q. On gap functions for vector variational inequalities. In: F. Giannessi Vector Variational Inequalities and Vector Equilibria, Mathematical Theories (pp. 55-72), (Ed.), Kluwer, Dordrecht (2000)

    Chapter  Google Scholar 

  8. Goh, C.J., Yang, X.Q. Duality in Optimization and Variational Inequalities, Taylor and Francis, London (2002)

    Book  MATH  Google Scholar 

  9. Sawaragi, Y., Nakayama, H., Tanino, T. Theory of Multiobjective Optimization, Mathematics in Science and Engineering, (Vol. 176), Academic, Orlando etc. (1985)

    Google Scholar 

  10. Song, W. Conjugate duality in set-valued vector optimization. J. Math. Anal. Appl. 216(1), 265–283 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Song, W. A generalization of Fenchel duality in set-valued vector optimization. Mathe Methods Oper. Res. 48(2) 259–272 (1998)

    Article  MATH  Google Scholar 

  12. Tanino, T., Sawaragi, Y. Conjugate maps and duality in multiobjective optimization. J. Optim. Theory Appl. 31, 473–499 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. Tanino, T. On supremum of a set in a multidimensional space. J. Math. Anal. Appl. 130(2), 386–397 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tanino, T. Conjugate duality in vector optimization. J. Math. Anal. Appl. 167(1), 84–97 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wanka, G., Boţ, R.I. On the relations between different dual problems in convex mathematical programming. In: P. Chamoni, R. Leisten, A. Martin, J. Minnemann, H. Stadler (Eds.), Operations Research Proceedings 2001 (pp. 255–262), Springer, Berlin, (2002)

    Google Scholar 

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Acknowledgments

The research of the first author has been supported partially by Deutsche Forschungsgemeinschaft. The authors are grateful to Dr. Radu Ioan Boţ for valuable discussions.

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Correspondence to Lkhamsuren Altangerel .

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Altangerel, L., Wanka, G. (2010). Gap Functions for Vector Equilibrium Problems via Conjugate Duality. In: Chinchuluun, ., Pardalos, P., Enkhbat, R., Tseveendorj, I. (eds) Optimization and Optimal Control. Springer Optimization and Its Applications(), vol 39. Springer, New York, NY. https://doi.org/10.1007/978-0-387-89496-6_10

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