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On Hölder Calmness and Hölder Well-Posedness of Vector Quasi-Equilibrium Problems

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Abstract

We study stability and well-posedness of parametric vector quasi-equilibrium problems. Weak continuity of orders not greater than one around a given point, in the sense of Hölder calmness of such orders, of solution maps is under consideration. Namely, we consider stability in terms of Hölder calmness of solution maps at the considered point of parameter. Sufficient conditions for such Hölder calmness are established for weak and strong vector quasi-equilibrium problems. When applied to the particular case of scalar equilibrium problems, our results recover recent ones appearing online first in the literature. Then we propose a Hölder well-posedness notion for parametric vector quasi-equilibrium problems, based on Hölder calmness of approximate solution maps, and derive sufficient conditions for Hölder well-posedness of both the mentioned weak and strong vector quasi-equilibrium problems.

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References

  1. Anh, L.Q., Khanh, P.Q.: On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems. J. Math. Anal. Appl. 321, 308–315 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anh, L.Q., Khanh, P.Q.: Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces. J. Glob. Optim. 37, 449–465 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anh, L.Q., Khanh, P.Q.: On the stability of the solution sets of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl. 135, 271–284 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Anh, L.Q., Khanh, P.Q.: Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions. J. Glob. Optim. 42, 515–531 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Anh, L.Q., Khanh, P.Q.: Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks I: upper semicontinuities. Set-Valued Anal. 16, 267–279 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Anh, L.Q., Khanh, P.Q.: Hölder continuity of the unique solution to quasiequilibrium problems in metric spaces. J. Optim. Theory Appl. 141, 37–54 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Anh, L.Q., Khanh, P.Q.: Sensitivity analysis for weak and strong vector quasiequilibrium problems. Vietnam J. Math. 37, 1–17 (2009)

    MathSciNet  Google Scholar 

  8. Anh, L.Q., Khanh, P.Q.: Continuity of solution maps of parametric quasiequilibrium problems. J. Glob. Optim. 46, 247–259 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Anh, L.Q., Khanh, P.Q., Tam, T.N.: On Hölder continuity of approximate solutions to parametric equilibrium problems. Nonlinear Anal. 75, 2293–2303 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Anh, L.Q., Kruger, A.Y., Thao, N.H.: On Hölder calmness of solution mappings in parametric equilibrium problems. Top. doi:10.1007/s11750-012-0259-3 (2012)

  11. Bianchi, M., Pini, R.: A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31, 445–450 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bianchi, M., Pini, R.: Sensitivity for parametric vector equilibria. Optimization 55, 221–230 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MATH  MathSciNet  Google Scholar 

  16. Chuong, T.D., Kruger, A.Y., Yao, J.C.: Calmness of efficient solution maps in parametric vector optimization. J. Glob. Optim. 51, 677–688 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, New York (2009)

    Book  MATH  Google Scholar 

  18. Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III, pp. 103–113. Academic Press, New York (1972)

    Google Scholar 

  19. Hai, N.X., Khanh, P.Q.: Existence of solutions to general quasiequilibrium problems and applications. J. Optim. Theory Appl. 133, 317–327 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hai, N.X., Khanh, P.Q.: The solution existence of general variational inclusion problems. J. Math. Anal. Appl. 328, 1268–1277 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16, 199–227 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Khanh, P.Q., Luu, L.M.: Upper semicontinuity of the solution set to parametric vector quasivariational inequalities. J. Glob. Optim. 32, 569–580 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Levy, A.B.: Calm minima in parameterized finite-dimensional optimization. SIAM J. Optim. 11, 160–178 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Li, S.J., Li, X.B., Wang, L.N., Teo, K.L.: The Hölder continuity of solutions to generalized vector equilibrium problems. Eur. J. Oper. Res. 199, 334–338 (2009)

    Article  MATH  Google Scholar 

  27. Mansour, M.A., Riahi, H.: Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 306, 684–691 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Mansour, M.A., Scrimali, L.: Hölder continuity of solutions to elastic traffic network models. J. Glob. Optim. 40, 175–184 (2008)

    Article  MATH  Google Scholar 

  29. Oettli, W., Yen, N.D.: Quasicomplementarity problems of type R 0. J. Optim. Theory Appl. 89, 467–474 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  30. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, 3rd edn. Springer, Berlin (2009)

    MATH  Google Scholar 

  31. Yen, N.D.: Hölder continuity of solutions to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zheng, X.Y., Ng, K.F.: Calmness for L-subsmooth multifunctions in Banach spaces. SIAM J. Optim. 19, 1648–1673 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

This research was funded by Vietnam National University Hochiminh City (VNU-HCM) under grant number B2013-28-01. The authors would like to thank the anonymous referee for his/her valuable remarks and suggestions, which have helped them to improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam

    Lam Quoc Anh

  2. Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Vietnam

    Phan Quoc Khanh

  3. Department of Mathematics, Taydo University, Cantho, Vietnam

    Tran Ngoc Tam

  4. Department of Mathematics, Cantho College, Cantho City, Vietnam

    Dang Thi My Van

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  1. Lam Quoc Anh
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  2. Phan Quoc Khanh
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  3. Tran Ngoc Tam
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  4. Dang Thi My Van
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Correspondence to Phan Quoc Khanh.

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Anh, L.Q., Khanh, P.Q., Tam, T.N. et al. On Hölder Calmness and Hölder Well-Posedness of Vector Quasi-Equilibrium Problems. Viet J Math 41, 507–517 (2013). https://doi.org/10.1007/s10013-013-0039-x

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  • DOI: https://doi.org/10.1007/s10013-013-0039-x

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