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Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems

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Abstract

In this paper, we introduce the concepts of globally efficient solution and cone-Benson efficient solution for a vector equilibrium problem; we give some scalarization results for Henig efficient solution sets, globally efficient solution sets, weak efficient solution sets, and cone-Benson efficient solution sets in locally convex spaces. Using the scalarization results, we show the connectedness and path connectedness of weak efficient solution sets and various proper efficient solution sets of vector equilibrium problem.

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References

  1. Ansari, Q.H., Oettli, W., Schläger, D.: A generalization of vector equilibria. Math. Methods Oper. Res. 46, 147–1527 (1997)

    Article  MATH  Google Scholar 

  2. Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997)

    Article  MATH  Google Scholar 

  3. Giannessi, F.: Theorem of the alternative, quadratic programs, and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, New York (1980)

    Google Scholar 

  4. Chen, G.Y., Cheng, G.M.: Vector variational inequalities and vector optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 258, pp. 408–416. Springer, Heidelberg (1987)

  5. Chen, G.Y., Yang, X.Q.: Vector complementarity problem and its equivalence with weak minimal element in ordered spaces. J. Math. Anal. Appl. 153, 136–158 (1990)

    Article  MATH  Google Scholar 

  6. Chen, G.Y.: Existence of solution for a vector variational inequality: an extension of the Hartman-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992)

    Article  MATH  Google Scholar 

  7. Yang, X.Q.: Vector variational inequality and its duality. Nonlinear Anal. Theory Methods Appl. 21, 869–877 (1993)

    Article  MATH  Google Scholar 

  8. Siddiqi, A.H., Ansari, Q.H., Khaliq, A.: On vector variational inequalities. J. Optim. Theory Appl. 84, 171–180 (1995)

    Article  MATH  Google Scholar 

  9. Chen, G.Y., Li, S.J.: Existence of solution for a generalized vector quasivariational inequality. J. Optim. Theory Appl. 90, 321–334 (1996)

    Article  MATH  Google Scholar 

  10. Yu, S.J., Yao, J.C.: On vector variational inequalities. J. Optim. Theory Appl. 89, 749–769 (1996)

    Article  MATH  Google Scholar 

  11. Lee, G.M., Lee, B.S., Chang, S.S.: On vector quasivariational inequalities. J. Math. Anal. Appl. 203, 626–638 (1996)

    Article  MATH  Google Scholar 

  12. Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42–58 (1997)

    Article  MATH  Google Scholar 

  13. Luo, Q.: Generalized vector variational-like inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 363–369. Kluwer, Dordrecht (2000)

    Google Scholar 

  14. Lee, G.M., Kim, D.S., Lee, B.S., Yun, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. Theory Methods Appl. 34, 745–765 (1998)

    Article  MATH  Google Scholar 

  15. Cheng, Y.H.: On the connectedness of the solution set for the weak vector variational inequality. J. Math. Anal. Appl. 260, 1–5 (2001)

    Article  MATH  Google Scholar 

  16. Gong, X.H.: Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl. 108, 139–154 (2001)

    Article  MATH  Google Scholar 

  17. Gong, X.H., Fu, W.T., Liu, W.: Superefficiency for a vector equilibrium in locally convex topological vector spaces. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 233–252. Kluwer, Dordrecht (2000)

    Google Scholar 

  18. Fu, J.Y.: Generalized vector quasiequilibrium problems. Math. Methods Oper. Res. 52, 57–64 (2000)

    Article  MATH  Google Scholar 

  19. Song, W.: Vector equilibrium problems with set-valued mapping. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 403–418. Kluwer, Dordrecht (2000)

    Google Scholar 

  20. Fang, Y.P., Huang, N.J.: Vector equilibrium type problems with (S)+-conditions. Optimization 53, 269–279 (2004)

    Article  MATH  Google Scholar 

  21. Chiang, C., Chadli, O., Yao, J.C.: Generalized vector equilibrium problems with trifunctions. J. Glob. Optim. 30, 135–154 (2004)

    Article  MATH  Google Scholar 

  22. Ding, X.P., Park, J.Y.: Generalized vector equilibrium problems in generalized convex spaces. J. Optim. Theory Appl. 120, 327–353 (2004)

    Article  MATH  Google Scholar 

  23. Lin, L.J., Ansari, Q.H., Wu, J.Y.: Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems. J. Optim. Theory Appl. 117, 121–137 (2003)

    Article  MATH  Google Scholar 

  24. Fu, J.Y.: Vector equilibrium problems. Existence theorems and convexity of the solution set. J. Glob. Optim. 31, 109–119 (2005)

    Article  MATH  Google Scholar 

  25. Zheng, X.Y.: The domination property for efficiency in locally convex spaces. J. Math. Anal. Appl. 213, 455–467 (1997)

    Article  MATH  Google Scholar 

  26. Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)

    Article  MATH  Google Scholar 

  27. Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)

    Article  MATH  Google Scholar 

  28. Borwein, J.M., Zhuang, D.M.: Superefficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)

    Article  MATH  Google Scholar 

  29. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Wiley, New York (1984)

    MATH  Google Scholar 

  30. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  31. Warburton, A.R.: Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives. J. Optim. Theory Appl. 40, 537–557 (1983)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanchang University, Nanchang, 330047, China

    X. H. Gong

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  1. X. H. Gong
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Correspondence to X. H. Gong.

Additional information

Communicated by H.P. Benson.

This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jinxing Province, China.

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Gong, X.H. Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems. J Optim Theory Appl 133, 151–161 (2007). https://doi.org/10.1007/s10957-007-9196-y

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