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Optimality conditions for the Henig efficient solution of vector equilibrium problems with constraints

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Abstract

The purpose of this paper is to establish optimality conditions for vector equilibrium problems with constraints. By using the separation of convex sets, we obtain the necessary and sufficient conditions for the Henig efficient solution and the superefficient solution to the vector equilibrium problem with constraints. As applications of our results, we derive some optimality conditions to the vector variational inequality problem and the vector optimization problem with constraints.

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References

  1. Anh L.Q., Khanh P.Q., Van D.T.M., Yao J.C.: Well-posedness for vector quasiequilibria. Taiwanese J. Math. 13, 713–737 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Ansari Q.H., Yao J.C.: An existence result for the generalized vector equilibrium. Appl. Math. Lett. 12, 53–56 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen G.Y., Yang X.Q., Yu H.: A nonlinear scalarization function and generalized quasi-vector equilibrium problem. J. Global Optim. 32, 451–466 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds): Pareto Optimality, Game Theory and Equilibria. Springer, Berlin (2008)

    MATH  Google Scholar 

  9. Floudas, C.A., Pardalos, P.M. (eds): Encyclopedia of Optimization. Springer, Berlin (2009)

    MATH  Google Scholar 

  10. Fu J.Y.: Generalized vector quasi-equilibrium problems. Math. Meth. Oper. Res. 52, 57–64 (2000)

    Article  MATH  Google Scholar 

  11. Giannessi F.: Theorem of alternative, quadratic programs, and complementarity problem. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds) Variational Inequality and Complementarity Problem., pp. 151–186. Wiley, Chichester (1980)

    Google Scholar 

  12. Giannessi, F. (ed.): Vector Variational Inequilities and Vector Equilibria: Mathematical Theories. Kluwer, Dordrechet (2000)

    Google Scholar 

  13. Giannessi F., Mastroeni G., Pellegrini L.: On the theory of vector optimization and variational inequalities. Image space analysis and separation. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 153–215. Kluwer, Dordrecht (2000)

    Chapter  Google Scholar 

  14. Giannessi, F., Maugeri, A., Pardalos, P.M. (eds): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Series: Nonconvex Optimization and Its Applications, vol. 58 (2002)

  15. Gong X.H.: Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307, 12–31 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gong X.H.: Optimality conditions for vector equilibrium problems. J. Math. Anal. Appl. 342, 1455–1466 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Gong X.H., Fu W.T., Liu W.: Super efficiency for 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)

    Chapter  Google Scholar 

  18. Gong X.H., Yao J.C.: Connectedness of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 189–196 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gong X.H., Yao J.C.: Lower semicontinuity of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 197–205 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hadjisavvas N., Schaibles S.: From scalar to vector equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 96, 297–309 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huang N.J., Fang Y.P.: On vector variational inequalities in reflexive Banach space. J. Global Optim. 32, 495–505 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huang N.J., Long X.J., Zhao C.W.: Well-posedness for vector quasi-equilibrium problems with applications. J. Ind. Manag. Optim. 5, 341–349 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kimura K., Yao J.C.: Sensitivity analysis of vector equilibrium problems. Taiwanese J. Math. 12, 649–669 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Kimura K., Yao J.C.: Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems. Taiwanese J. Math. 12, 2233–2268 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Kimura K., Yao J.C.: Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems. J. Global Optim. 41, 187–202 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kimura K., Yao J.C.: Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems. J. Ind. Manag. Optim. 4, 167–181 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Long X.J., Huang N.J.: Gap functions and existence of solutions for generalized vector quasi-variational inequalities. Involving 1, 183–195 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Long X.J., Huang N.J., Teo K.L.: Existence and stability of solutions for generalized strong vector quasi-equilibrium problem. Math. Comput. Modelling 47, 445–451 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Long X.J., Xiang Z.H., Min X.C.: Some existence results of solutions for generalized vector variational inequalities in Banach spaces. Adv. Nonlinear Variational Inequalities 10, 143–150 (2007)

    MathSciNet  Google Scholar 

  31. Luc, D.T.: Theory of Vector Optimization, Lecture Notes in Economics and Mathematics Systems, vol. 319. Springer, New York (1989)

  32. Morgan J., Romaniello M.: Scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities. J. Optim. Theory Appl. 130, 309–316 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pardalos, P.M., Resende, M. (eds): Handbook of Applied Optimization. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  34. Qiu Q.S.: Optimality conditions for vector equilibrium problems with constraints. J. Ind. Manag. Optim. 5, 783–790 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Song W.: On generalized vector equilibrium problems. J. Comput. Appl. Math. 146, 167–177 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yang X.M., Li D., Wang S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413–427 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yang X.Q., Yao J.C.: Gap functions and existence of solutions to set-valued vector variational inequalities. J. Optim. Theory Appl. 115, 407–417 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People’s Republic of China

    X. J. Long & Y. Q. Huang

  2. College of Science, Chongqing JiaoTong University, Chongqing, 400074, People’s Republic of China

    Z. Y. Peng

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  1. X. J. Long
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  2. Y. Q. Huang
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  3. Z. Y. Peng
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Long, X.J., Huang, Y.Q. & Peng, Z.Y. Optimality conditions for the Henig efficient solution of vector equilibrium problems with constraints. Optim Lett 5, 717–728 (2011). https://doi.org/10.1007/s11590-010-0241-7

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  • DOI: https://doi.org/10.1007/s11590-010-0241-7

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