Variational Inequality Theory

  • Anna Nagurney
Part of the Advances in Computational Economics book series (AICE, volume 10)


Equilibrium is a concept central to the analysis of economic phenomena. Methodologies that have been applied to the formulation, qualitative analysis, and computation of economic equilibria have included systems of equations, optimization theory, complementarity theory, as well as fixed point theory. In this chapter the foundations for the theory of variational inequalities are established and the relationship of this methodology to other existing equilibrium analysis tools identified. Variational inequality theory will be utilized throughout the book as the fundamental methodology in synthesizing network economic equilibrium models operating under a spectrum of behavioral mechanisms and ranging from spatial price equilibrium problems and imperfectly competitive oligopolistic market equilibrium problems to general financial equilibrium problems.


Equilibrium Point Variational Inequality Equilibrium Problem Complementarity Problem Variational Inequality Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bershchanskii, Y. M., and Meerov, M. V., “The complementarity problem: theory and methods of solution,” Automation and Remote Control 44 (1983) 687–710.Google Scholar
  2. Border, K. C., Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, Cambridge, United Kingdom, 1985.Google Scholar
  3. Dafermos, S., “Traffic equilibria and variational inequalities,” Trans-portation Science 14 (1980) 42–54.CrossRefGoogle Scholar
  4. Dafermos, S., “Sensitivity analysis in variational inequalities,” Mathematics of Operations Research 13 (1988) 421–434.CrossRefGoogle Scholar
  5. Dafermos, S. C., and McKelvey, S. C., “Partitionable variational inequalities with applications to network and economic equilibria,” Journal of Optimization Theory and Applications 73 (1992) 243–268.CrossRefGoogle Scholar
  6. Dafermos, S., and Nagurney, A., “Sensitivity analysis for the asymmetric network equilibrium problem,” Mathematical Programming 28 (1984a) 174–184.CrossRefGoogle Scholar
  7. Dafermos, S., and Nagurney, A., “Sensitivity analysis for the general spatial economic equilibrium problem,” Operations Research 32 (1984b) 1069–1086.CrossRefGoogle Scholar
  8. Dupuis, P., and Ishii, H., “On Lipschitz continuity of the solution mapping to the Skorokhod Problem, with applications,” Stochastic and Stochastic Reports 35 (1991) 31–62.Google Scholar
  9. Dupuis, P., and Nagurney, A., “Dynamical systems and variational inequalities,” Annals of Operations Research 44 (1993) 9–42.CrossRefGoogle Scholar
  10. Hartman, P., and Stampacchia, G., “On some nonlinear elliptic differential functional equations,” Acta Mathematica 115 (1966) 271–310.CrossRefGoogle Scholar
  11. Karamardian, S., “The nonlinear complementarity problem with applications, part 1,” Journal of Optimization Theory and Applications 4 (1969) 87–98.CrossRefGoogle Scholar
  12. Kelley, J. L., General Topology, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1955.Google Scholar
  13. Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.Google Scholar
  14. Kostreva, M. M., “Recent results on complementarity models for engineering and economics,” INFOR 28 (1990) 324–334.Google Scholar
  15. Kyparisis, J., “Sensitivity analysis framework for variational inequalities,” Mathematical Programming 38 (1987) 203–213.CrossRefGoogle Scholar
  16. Lemke, C. E., “Recent results on complementarity problems,” in Nonlinear Programming, pp. 349–384, J. B. Rosen, O. L. Mangasarian, and K. Ritter, editors, Academic Press, New York, 1970.Google Scholar
  17. Lemke, C. E., “A survey of complementarity problems,” in Variational Inequalities and Complementarity Problems, pp. 213–239, R. W. Cottle, F. Giannessi, and J. L. Lions, editors, John Wiley Sons, Chichester, England, 1980.Google Scholar
  18. Mancino, O., and Stampacchia, G., “Convex programming and variational inequalities,” Journal of Optimization Theory and Applications 9 (1972) 3–23.CrossRefGoogle Scholar
  19. McKelvey, S. C., “Partitionable variational inequalities and an application to the market equilibrium problem,” Ph. D. thesis, Division of Applied Mathematics, Brown University, Providence, Rhode Island, 1989.Google Scholar
  20. Nagurney, A., editor, Advances in Equilibrium Modeling, Analysis, and Computation, Annals of Operations Research 44, J. C. Baltzer AG Scientific Publishing Company, Basel, Switzerland, 1993.Google Scholar
  21. Nagurney, A., and Zhang, D., Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers, Boston, Massachusetts, 1996.Google Scholar
  22. Qiu, Y., and Magnanti, T. L., “Sensitivity analysis for variational inequalities,” Mathematics of Operations Research 17 (1992) 61–70.CrossRefGoogle Scholar
  23. Robinson, S. M., “Strongly regular generalized equations,” Mathematics of Operations Research 5 (1980) 43–62.CrossRefGoogle Scholar
  24. Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  25. Smith, M. J., “Existence, uniqueness, and stability of traffic equilibria,” Transportation Research 13B (1979) 295–304.Google Scholar
  26. Tobin, R. L., “Sensitivity analysis for variational inequalities,” Journal of Optimization Theory and Applications 48 (1986) 191–204.Google Scholar
  27. Zhang, D., and Nagurney, A., “On the stability of projected dynamical systems,” Journal of Optimization Theory and Applications 85 (1995) 97–124.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Anna Nagurney
    • 1
  1. 1.University of MassachusettsAmherstUSA

Personalised recommendations