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On Essential Stable Sets of Solutions in Set Optimization Problems

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Abstract

This paper introduces the essential stability for set optimization problems. Some kinds of essential stable sets of weakly minimal and minimal solutions are shown. The graph of minimal solution mappings is not necessarily closed, which is different from weakly minimal solution mappings. The existence of minimum essential sets of minimal solutions is proved.

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References

  1. Fort, M.K.: Essential and nonessential fixed points. Am. J. Math. 72, 315–322 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  2. Yu, J., Xiang, S.W.: The stability of the set of KKM points. Nonlinear Anal. 54, 839–844 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Khanh, P.Q., Quan, N.H.: Generic stability and essential components of generalized KKM points and applications. J. Optim. Theory Appl. 148, 488–504 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Yang, H., Yu, J.: On essential components of the set of weakly pareto-Nash equilibrium points. Appl. Math. Lett. 15, 553–560 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carbonell-Nicolau, O.: Essential equilibria in normal-form games. J. Econ. Theory 145, 421–431 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Luo, Q.: Essential component and essential optimum solution of optimization problems. J. Optim. Theory Appl. 102, 433–438 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Song, Q.Q., Wang, L.S.: On the stability of the solution for multiobjective generalized games with the payoffs perturbed. Nonlinear Anal. 73, 2680–2685 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlinear Anal. 71, 3769–3778 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Xiang, S.W., Zhou, Y.H.: On essential sets and essential components of efficient solutions for vector optimization problems. J. Math. Anal. Appl. 315, 317–326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Luc, D.T.: Quasiconcave vector maximization: connectedness of the sets of pareto-optimal and weak pareto-optimal alternatives. J. Optim. Theory Appl. 122, 346–354 (1987)

    MathSciNet  MATH  Google Scholar 

  11. Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Berlin (2004)

    MATH  Google Scholar 

  12. Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Econom. and Math. Systems, vol. 319. Springer, Berlin (1989)

    Book  Google Scholar 

  13. Giannessi, F.: Constrained Optimization and Image Space Analysis. Separation of Sets and Optimality Conditions, vol. 1. Springer, New York (2005)

    MATH  Google Scholar 

  14. Chinaie, M., Zafarani, J.: Image space analysis and scalarization of set-valued optimization. J. Optim. Theory Appl. 142, 451–467 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lalitha, C.S., Dutta, J., Govil, M.G.: Optimality criteria in set-valued optimization. J. Aust. Math. Soc. 75, 221–231 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ruchi, A., Lalitha, C.S.: Proximal proper efficiency in set-valued optimization. Omega 33, 407–411 (2005)

    Article  Google Scholar 

  17. Khanh, P.Q., Quy, D.N.: On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings. J. Glob. Optim. 49, 381–396 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jahn, J., Ha, T.: New order relations in set optimization. J. Optim. Theory Appl. 148, 209–236 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rodríguez-Marín, L., Sama, G.: (Λ,C)-contingent derivatives of set-valued maps. J. Math. Anal. Appl. 335(2), 974–989 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Löhne, A.: Optimization with set relations: conjugate duality. Optimization 54(3), 265–282 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Alonso, M., Rodríguez-Marín, L.: Set-relations and optimality conditions in set-valued maps. Nonlinear Anal. 63(8), 1167–1179 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kuroiwa, D.: Some duality theorems of set-valued optimization with natural criteria. In: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, pp. 221–228. World Scientific, River Edge (1999)

    Google Scholar 

  24. Yu, J., Yang, H., Xiang, S.W.: Unified approach to existence and stability of essential components. Nonlinear Anal. TMA 63, e2415–e2425 (2006)

    Google Scholar 

  25. Yu, J., Zhou, Y.: A Hausdorff metric inequality with applications to the existence of essential components. Nonlinear Anal. TMA 69, 1851–1855 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank all reviewers for helpful comments and thank editors for useful suggestions.

This project is supported by Natural Science Foundation of Guangxi, China (No. 2012GXNSFBA053013) and Guangxi Key Laboratory of Spatial Information and Geomatics (1103108-24).

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Authors and Affiliations

  1. College of Science, Guilin University of Technology, Guilin, 541004, China

    Q. Q. Song & G. Q. Tang

  2. College of Science, China Agricultural University, Beijing, 100083, China

    L. S. Wang

Authors
  1. Q. Q. Song
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  2. G. Q. Tang
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  3. L. S. Wang
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Correspondence to Q. Q. Song.

Additional information

Communicated by Jafar Zafarani.

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Song, Q.Q., Tang, G.Q. & Wang, L.S. On Essential Stable Sets of Solutions in Set Optimization Problems. J Optim Theory Appl 156, 591–599 (2013). https://doi.org/10.1007/s10957-012-0129-z

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  • DOI: https://doi.org/10.1007/s10957-012-0129-z

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