Limit vector variational inequalities and market equilibrium problems

  • M. Bianchi
  • I. V. Konnov
  • R. PiniEmail author
Original Paper


In a finite-dimensional setting we investigate the solvability of a general vector variational inequality via the convergence of solutions of suitable approximating vector variational inequalities defined with more regular data. The theoretical results obtained in a very general framework are successfully applied to the study of a vector market equilibrium problem where instead of exact values of the cost mapping, feasible set and order cone, only approximation sequences of these data are available.


Vector variational inequality Kuratowski convergence Approximation sequence Coercivity conditions 



In this work, the second author was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, Project No. 1.13556.2019/13.1 and was also supported by Russian Foundation for Basic Research, Project No. 19-01-00431.


  1. 1.
    Giannessi, F.: Theorems of alternative, quadratic programs and complementarity problems. In: Variational Inequalities and Complementarity Problems (Proc. Internat. School, Erice, 1978), pp. 151–186. Wiley, Chichester (1980)Google Scholar
  2. 2.
    Daniilidis, A., Hadjisavvas, N.: Existence theorems for vector variational inequalities. Bull. Austral. Math. Soc. 54, 473–481 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Oettli, W.: A remark on vector-valued equilibria and generalized monotonicity. Acta Math. Vietnam. 22, 213–221 (1997)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Kluwer Academic Publishers, Dordrecht (2000)zbMATHGoogle Scholar
  5. 5.
    Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization. Springer, Berlin (2005)Google Scholar
  6. 6.
    Konnov, I.V.: Application of penalty methods to non-stationary variational inequalities. Nonlinear Anal. Theory Methods Appl. 92, 177–182 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Konnov, I.V.: An inexact penalty method for non stationary generalized variational inequalities. Set-Valued Var. Anal. 23, 239–248 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bianchi, M., Konnov, I.V., Pini, R.: Barrier methods for equilibrium problems. Pure Appl. Funct. Anal. 2, 1–10 (2017)MathSciNetGoogle Scholar
  9. 9.
    Bianchi, M., Konnov, I.V., Pini, R.: Limit vector variational inequality problems via scalarization. J. Glob. Optim. 72, 579–590 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Konnov, I.V.: On vector formulations of auction-type problems with applications. Optimization 65, 233–251 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sukharev, A.G., Timokhov, A.V., Fedorov, V.V.: A Course in Optimization Methods. Nauka, Moscow (1986)zbMATHGoogle Scholar
  12. 12.
    Konnov, I.V.: On scalarization of vector optimization type problems. Russian Math. (Iz. VUZ) 56, 5–13 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lucchetti, R., Patrone, F.: Closure and upper semicontinuity results in mathematical programming. Optimization 5, 619–628 (1986)CrossRefGoogle Scholar
  14. 14.
    Konnov, I.V., Liu, Z.: Vector equilibrium problems on unbounded sets. Lobachevskii J. Math. 31, 232–238 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Konnov, I.V.: An alternative economic equilibrium model with different implementation mechanisms. Adv. Model. Optim. 17, 245–265 (2015)zbMATHGoogle Scholar
  16. 16.
    Konnov, I.V.: On auction equilibrium models with network applications. Netnomics 16, 107–125 (2015)CrossRefGoogle Scholar
  17. 17.
    Salinetti, G., Wets, R.J.-B.: O, the convergence of sequences of convex sets in finite dimensions. SIAM Rev. 21, 18–33 (1979)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica per le Scienze Economiche, Finanziarie ed AttuarialiUniversità Cattolica del Sacro CuoreMilanItaly
  2. 2.Department of System Analysis and Information TechnologiesKazan UniversityKazanRussia
  3. 3.Dipartimento di Matematica e ApplicazioniUniversità degli Studi Milano-BicoccaMilanItaly

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