Levitin–Polyak well-posedness for equilibrium problems with the lexicographic order

Abstract

The aim of this work is to investigate optimization-related problems with the objective spaces ordered by the lexicographic cones, including parametric lexicographic equilibrium problems and optimization problems with lexicographic equilibrium constraints. We introduce concepts of Levitin–Polyak well-posedness for these problems and establish a number of sufficient conditions for such properties. The assumptions are imposed directly on the data of the problems and really verifiable. We do not need to suppose the existence (and/or convexity, compactness) of the solution set because it is proved using the mentioned assumptions on the data. Moreover, our assumptions are more relaxed than those which are usually imposed.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. 1.

    Anh, L.Q., Duy, T.Q.: Tykhonov well-posedness for lexicographic equilibrium problems. Optimization 65(11), 1929–1948 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Anh, L.Q., Duy, T.Q.: On penalty method for equilibrium problems in lexicographic order. Positivity 22(1), 39–57 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Anh, L.Q., Duy, T.Q., Khanh, P.Q.: Continuity properties of solution maps of parametric lexicographic equilibrium problems. Positivity 20(1), 61–80 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Anh, L.Q., Duy, T.Q., Kruger, A.Y., Thao, N.H.: Well-posedness for lexicographic vector equilibrium problems. In: Demyanov, V.F. (ed.) Constructive Nonsmooth Analysis and Related Topics, vol. 87, pp. 159–174. Springer, New York (2014)

    Google Scholar 

  5. 5.

    Bianchi, M., Konnov, I.V., Pini, R.: Lexicographic variational inequalities with applications. Optimization 56(3), 355–367 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Bianchi, M., Konnov, I.V., Pini, R.: Lexicographic and sequential equilibrium problems. J. Glob. Optim. 46(4), 551–560 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Bianchi, M., Pini, R.: A note on equilibrium problems with properly quasimonotone bifunctions. J. Glob. Optim. 20(1), 67–76 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90(1), 31–43 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Caruso, F., Ceparano, M.C., Morgan, J.: Uniqueness of nash equilibrium in continuous two-player weighted potential games. J. Math. Anal. Appl. 459(2), 1208–1221 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Crespi, G.P., Papalia, M., Rocca, M.: Extended well-posedness of quasiconvex vector optimization problems. J. Optim. Theory Appl. 141(2), 285–297 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Crouzeix, J.P., Marcotte, P., Zhu, D.: Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities. Math. Program. 88(3), 521–539 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Darabi, M., Zafarani, J.: Levitin-Polyak well-posedness of strong parametric vector quasi-equilibrium problems. In: Cushing, J., Saleem, M., Srivastava, H., Khan, M., Merajuddin, M. (eds.) Applied Analysis in Biological and Physical Sciences, pp. 321–337. Springer, New Delhi (2016)

    Google Scholar 

  13. 13.

    Dempe, S.: Foundations of Bilevel Programming. Springer, Berlin (2002)

    Google Scholar 

  14. 14.

    Fang, Y.P., Hu, R., Huang, N.J.: Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints. Comput. Math. Appl. 55(1), 89–100 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75(4), 1822–1833 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Halkin, H.: Implicit functions and optimization problems without continuous differentiability of the data. SIAM J. Control 12(2), 229–236 (1974)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Hu, R., Fang, Y.P.: Characterizations of Levitin-Polyak well-posedness by perturbations for the split variational inequality problem. Optimization 65(9), 1717–1732 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Huang, X.X., Yang, X.Q.: Generalized Levitin-Polyak well-posedness in constrained optimization. SIAM J. Optim. 17(1), 243–258 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Ioffe, A., Lucchetti, R.E.: Typical convex program is very well posed. Math. Program. 104(2–3), 483–499 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2009)

    Google Scholar 

  21. 21.

    John, R.: The concave nontransitive consumer. J. Glob. Optim. 20(3–4), 297–308 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Khoshkhabar-amiranloo, S., Khorram, E.: Scalarization of Levitin-Polyak well-posed set optimization problems. Optimization 66(1), 113–127 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Kimura, K., Liou, Y.C., Wu, S.Y., Yao, J.C.: Well-posedness for parametric vector equilibrium problems with applications. J. Ind. Manag. Optim. 4(2), 313 (2008)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Kohli, R., Jedidi, K.: Representation and inference of lexicographic preference models and their variants. Marketing Science 26(3), 380–399 (2007)

    Article  Google Scholar 

  25. 25.

    Konnov, I.V.: On lexicographic vector equilibrium problems. J. Optim. Theory Appl. 118(3), 681–688 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Konnov, I.V., Schaible, S.: Duality for equilibrium problems under generalized monotonicity. J. Optim. Theory Appl. 104(2), 395–408 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Küçük, M., Soyertem, M., Küçük, Y.: On constructing total orders and solving vector optimization problems with total orders. J. Glob. Optim 50(2), 235–247 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Lalitha, C.S., Bhatia, G.: Levitin-Polyak well-posedness for parametric quasivariational inequality problem of the minty type. Positivity 16(3), 527–541 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Lalitha, C.S., Chatterjee, P.: Levitin-Polyak well-posedness for constrained quasiconvex vector optimization problems. J. Glob. Optim 59(1), 191–205 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Levitin, E.S., Polyak, B.T.: On the convergence of minimizing sequences in conditional extremum problems. Dokl. Akad. Nauk SSSR 168(5), 997–1000 (1966)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Li, S.J., Li, M.H.: Levitin-Polyak well-posedness of vector equilibrium problems. Math. Methods Oper. Res. 69(1), 125–140 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Lignola, M.B., Morgan, J.: Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. J. Glob. Optim 16(1), 57–67 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Lin, L.J., Du, W.S.: On maximal element theorems, variants of Ekeland’s variational principle and their applications. Nonlinear Anal. 68(5), 1246–1262 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Mäkelä, M.M., Nikulin, Y.: On cone characterizations of strong and lexicographic optimality in convex multiobjective optimization. J. Optim. Theory Appl. 143(3), 519–538 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Martinez-Legaz, J.E.: Lexicographic Utility and Orderings. Handbook of Utility Theory 1, 345–369 (1998)

    Google Scholar 

  36. 36.

    Mas-Colell, A., Whinston, M.D., Green, J.R., et al.: Microeconomic Theory, vol. 1. Oxford University, New York (1995)

    Google Scholar 

  37. 37.

    Maugeri, A., Raciti, F.: On existence theorems for monotone and nonmonotone variational inequalities. J. Convex Anal. 16(3–4), 899–911 (2009)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Miglierina, E., Molho, E.: Well-posedness and convexity in vector optimization. Math. Meth. Oper. Res. 58(3), 375–385 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126(2), 391–409 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Monderer, D., Shapley, L.S.: Potential games. Game Eco. Behavior 14(1), 124–143 (1996)

    Article  Google Scholar 

  41. 41.

    Morgan, J., Scalzo, V.: New results on value functions and applications to maxsup and maxinf problems. J. Math. Anal. Appl. 300(1), 68–78 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Morgan, J., Scalzo, V.: Pseudocontinuity in optimization and nonzero-sum games. J. Optim. Theory Appl. 120(1), 181–197 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Morgan, J., Scalzo, V.: Discontinuous but well-posed optimization problems. SIAM J. Optim. 17(3), 861–870 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Shafer, W.J.: The nontransitive consumer. Econometrica 42(5), 913–919 (1974)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Tykhonov, A.N.: On the stability of the functional optimization problem. USSR Comput. Math. Math. Phys. 6(4), 28–33 (1966)

    Article  Google Scholar 

  46. 46.

    Virmani, G., Srivastava, M.: On Levitin-Polyak \(\alpha \)-well-posedness of perturbed variational-hemivariational inequality. Optimization 64(5), 1153–1172 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Wang, G., Huang, X.X.: Levitin-Polyak well-posedness for optimization problems with generalized equilibrium constraints. J. Optim. Theory Appl. 153(1), 27–41 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Wangkeeree, R., Bantaojai, T.: Levitin-Polyak well-posedness for lexicographic vector equilibrium problems. J. Nonlinear Sci. Appl. 10(2), 354–367 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Wangkeeree, R., Bantaojai, T., Yimmuang, P.: Well-posedness for lexicographic vector quasiequilibrium problems with lexicographic equilibrium constraints. J. Inequal. Appl. 163, 1–24 (2015)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Zolezzi, T.: Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal. 25(5), 437–453 (1995)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the Editors and anonymous Referees for their helpful remarks and suggestions that helped us significantly improve the paper. The second author is supported by the Institute for Computational Science and Technology (ICST) under the grant number 03/2019/HD-KHCNTT.

Author information

Affiliations

Authors

Corresponding author

Correspondence to T. Q. Duy.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Anh, L.Q., Duy, T.Q. & Khanh, P.Q. Levitin–Polyak well-posedness for equilibrium problems with the lexicographic order. Positivity (2021). https://doi.org/10.1007/s11117-021-00818-5

Download citation

Keywords

  • Lexicographic equilibrium problem
  • Optimization problem with lexicographic equilibrium constraints
  • Levitin–Polyak well-posedness
  • Potential game
  • Nontransitive consumer

Mathematics Subject Classification

  • 49K40
  • 90C31
  • 91B50