, Volume 22, Issue 1, pp 39–57 | Cite as

On penalty method for equilibrium problems in lexicographic order

  • L. Q. Anh
  • T. Q. Duy


In this paper, we consider lexicographic vector equilibrium problems. We propose a penalty function method for solving such problems. We show that every penalty trajectory of the penalized lexicographic equilibrium problem tends to the solution of the original problem. Using the regularized gap function to obtain an error bound result for such penalized problems is given.


Lexicographic order Equilibrium problem Penalty method Gap function Error bound 

Mathematics Subject Classification

47J20 49M37 90C30 



The authors wish to thank the anonymous referees for the careful reviews and valuable comments that helped us significantly improve the presentation of the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2017.18.


  1. 1.
    Anh, L.Q., Duy, T.Q., Khanh, P.Q.: Continuity properties of solution maps of parametric lexicographic equilibrium problems. Positivity 20, 61–80 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anh, L.Q., Duy, T.Q., Kruger, A.Y., Thao, N.H.: Well-posedness for lexicographic vector equilibrium problems. In: Constructive Nonsmooth Analysis and Related Topics, pp. 159–174. Springer (2014)Google Scholar
  3. 3.
    Auchmuty, G.: Variational principles for variational inequalities. Numer. Funct. Anal. Optim. 10, 863–874 (1989)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Auslender, A.: Optimisation Méthodes Numériques. Masson, Paris (1976)MATHGoogle Scholar
  5. 5.
    Auslender, A.: Asymptotic analysis for penalty and barrier methods in variational inequalities. SIAM J. Control Optim. 37, 653–671 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Berge, C.: Topological Spaces. Oliver and Boyd, London (1963)MATHGoogle Scholar
  7. 7.
    Bianchi, M., Konnov, I.V., Pini, R.: Lexicographic variational inequalities with applications. Optimization 56, 355–367 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bianchi, M., Konnov, I.V., Pini, R.: Lexicographic and sequential equilibrium problems. J. Glob. Optim. 46, 551–560 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bigi, G., Castellani, M., Pappalardo, M.: A new solution method for equilibrium problems. Optim. Methods Softw. 24, 895–911 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bigi, G., Passacantando, M.: Twelve monotonicity conditions arising from algorithms for equilibrium problems. Optim. Methods Softw. 30, 323–337 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Am. Math. Soc. 49, 1–23 (1943)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Crouzeix, J.P., Marcotte, P., Zhu, D.: Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities. Math. Program. 88, 521–539 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dempe, S.: Foundations of Bilevel Programming. Springer, Berlin (2002)MATHGoogle Scholar
  15. 15.
    Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM J. Control Optim. 27, 1333–1360 (1989)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dinh, B.V., Muu, L.D.: On penalty and gap function methods for bilevel equilibrium problems. J. Appl. Math. 2011, 1–14 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Facchinei, F., Kanzow, C.: Penalty methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 20, 2228–2253 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Facchinei, F., Pang, J.S.: Exact penalty functions for generalized Nash problems. In: Large-Scale Nonlinear Optimization, pp. 115–126. Springer (2006)Google Scholar
  19. 19.
    Fishburn, P.C.: Exceptional paper-lexicographic orders, utilities and decision rules: a survey. Manag. Sci. 20, 1442–1471 (1974)CrossRefMATHGoogle Scholar
  20. 20.
    Frisch, R.: Principles of linear programming with particular reference to the double gradient form of the logarithmic potential method. Memorandum from the Institute of Economics, University of Oslo (1954)Google Scholar
  21. 21.
    Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Gupta, R., Mehra, A.: Gap functions and error bounds for quasi variational inequalities. J. Glob. Optim. 53, 737–748 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Gwinner, J.: On the penalty method for constrained variational inequalities. Optim. Theory Algorithms 86, 197–211 (1981)MathSciNetGoogle Scholar
  24. 24.
    Huang, L.R., Ng, K.F.: Equivalent optimization formulations and error bounds for variational inequality problems. J. Optim. Theory Appl. 125, 299–314 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Khan, S.A., Chen, J.W.: Gap functions and error bounds for generalized mixed vector equilibrium problems. J. Optim. Theory Appl. 166, 767–776 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Konnov, I.V.: On lexicographic vector equilibrium problems. J. Optim. Theory Appl. 118, 681–688 (2003)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Konnov, I.V.: Regularized penalty method for general equilibrium problems in banach spaces. J. Optim. Theory Appl. 164, 500–513 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Konnov, I.V., Pinyagina, O.V.: D-gap functions for a class of equilibrium problems in Banach spaces. Comput. Methods Appl. Math. 3, 274–286 (2003)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Li, S., Teo, K.L., Yang, X., Wu, S.Y.: Gap functions and existence of solutions to generalized vector quasi-equilibrium problems. J. Glob. Optim. 34, 427–440 (2006)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Mäkelä, M.M., Nikulin, Y.: On cone characterizations of strong and lexicographic optimality in convex multiobjective optimization. J. Optim. Theory Appl. 143, 519–538 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Marcotte, P., Zhu, D.L.: Exact and inexact penalty methods for the generalized bilevel programming problem. Math. Program. 74, 141–157 (1996)MathSciNetMATHGoogle Scholar
  32. 32.
    Martinez-Legaz, J.: Lexicographical order, inequality systems and optimization. In: System Modelling and Optimization, pp. 203–212. Springer (1984)Google Scholar
  33. 33.
    Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27, 411–426 (2003)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Moudafi, A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. 47, 287–292 (2010)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Muu, L.D., Oettli, W.: A lagrangian penalty function method for monotone variational inequalities. Numer. Funct. Anal. Optim. 10, 1003–1017 (1989)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model. J. Optim. Theory Appl. 142, 185–204 (2009)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving equilibrium problems. Math. Program. 116, 529–552 (2009)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Penot, J.P.: Calculus Without Derivatives. Springer, Berlin (2012)MATHGoogle Scholar
  39. 39.
    Peypouquet, J.: Convex Optimization in Normed Spaces: Theory, Methods and Examples. Springer, Berlin (2015)CrossRefMATHGoogle Scholar
  40. 40.
    Pinar, M.Ç., Zenios, S.A.: On smoothing exact penalty functions for convex constrained optimization. SIAM J. Optim. 4, 486–511 (1994)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Rubin, H., Ungar, P.: Motion under a strong constraining force. Commun. Pure Appl. Math. 10, 65–87 (1957)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Yeniay, O.: Penalty function methods for constrained optimization with genetic algorithms. Math. Comput. Appl. 10, 45–56 (2005)Google Scholar

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Teacher CollegeCantho UniversityCanthoVietnam
  2. 2.Department of MathematicsCantho Technical and Economic CollegeCanthoVietnam
  3. 3.Department of MathematicsUniversity of Science, Vietnam National University Ho Chi Minh CityHo Chi Minh CityVietnam

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