Annals of Operations Research

, Volume 44, Issue 1, pp 7–42 | Cite as

Dynamical systems and variational inequalities

  • Paul Dupuis
  • Anna Nagurney
Methodological Advances


The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.


Dynamical systems variational inequalities the Skorokhod Problem equilibrium problems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.P. Aubin and A. Cellina,Differential Inclusions (Springer, Berlin, 1984).Google Scholar
  2. [2]
    M. Beckmann, C.B. McGuire and C.B. Winsten,Studies in the Economics of Transportation (Yale University Press, New Haven, CT, 1956).Google Scholar
  3. [3]
    D. Bernstein, Programmability of continuous and discrete network equilibria, PhD Thesis, University of Pennsylvania, Philadelphia, PA (1990).Google Scholar
  4. [4]
    P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).Google Scholar
  5. [5]
    K.C. Border,Fixed Point Theorems with Applications to Economics and Game Theory (Cambridge University Press, Cambridge, England, 1985).Google Scholar
  6. [6]
    E.A. Coddington and N. Levinson,Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).Google Scholar
  7. [7]
    A.A. Cournot,Researches into the Mathematical Principles of the Theory of Wealth (1838; English transl.: MacMillan, London, England, 1897).Google Scholar
  8. [8]
    S. Dafermos, Traffic equilibrium and variational inequalities, Transp. Sci. 14 (1980) 42–54.Google Scholar
  9. [9]
    S. Dafermos, The general multimodal traffic equilibrium problem with elastic demand, Networks 12 (1982) 57–72.Google Scholar
  10. [10]
    S. Dafermos, An iterative scheme for variational inequalities, Math. Progr. 26 (1983) 40–47.Google Scholar
  11. [11]
    S. Dafermos, Exchange price equilibria and variational inequalities, Math. Progr. 46 (1990) 391–402.Google Scholar
  12. [12]
    S. Dafermos and A. Nagurney, Oligopolistic and competitive behavior of spatially separated markets, Reg. Sci. Urban Econ. 17 (1987) 245–254.Google Scholar
  13. [13]
    P. Dupuis, Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets, Stochastics 21 (1987) 63–96.Google Scholar
  14. [14]
    P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod Problem, with applications, Stochastics and Stochastic Reports 35 (1991) 31–62.Google Scholar
  15. [15]
    P. Dupuis and H. Ishii, SDEs with oblique reflection on nonsmooth domains, Ann. Prob. 21 (1993) 554–580.Google Scholar
  16. [16]
    P. Dupuis and H.J. Kushner, Stochastic approximation and large deviations: Upper bounds and w.p. 1 convergence, SIAM J. Control Optim. 27 (1989) 1108–1135.Google Scholar
  17. [17]
    P. Dupuis and R.J. Williams, Lyapunov functions for semimartingale reflected Brownian motions, Lefschetz Center for Dynamical Systems, Report # 92-5, Brown University, Providence, RI (1992), to appear in: Ann. Prob.Google Scholar
  18. [18]
    S.N. Ethier and T.G. Kurtz,Markov Processes: Characterization and Convergence (Wiley, New York, 1986).Google Scholar
  19. [19]
    A.N. Fillipov, Differential equations with discontinuous right-hand side, Mat. Sbornik (N.S.) 51 (1960) 99–128.Google Scholar
  20. [20]
    S.D. Flam, Solving convex programs by means of ordinary differential equations, Math. Oper. Res. 17 (1992) 290–302.Google Scholar
  21. [21]
    S.D. Flam and A. Ben-Israel, A continuous approach to oligopolistic market equilibrium, Oper. Res. 38 (1990) 1045–1051.Google Scholar
  22. [22]
    M. Florian and M. Los, A new look at static spatial price equilibrium problems, Reg. Sci. Urban Econ. 12 (1982), 579–597.Google Scholar
  23. [23]
    M.I. Freidlin and A.D. Wentzell,Random Perturbations of Dynamical Systems (Springer, New York, 1984).Google Scholar
  24. [24]
    D. Gabay and H. Moulin, On the uniqueness and stability of Nash equilibria in noncooperative games, in:Applied Stochastic Control of Econometrics and Management Science, eds. A. Bensoussan, P. Kleindorfer and C.S. Tapiero (North-Holland, Amsterdam, The Netherlands, 1980).Google Scholar
  25. [25]
    C.B. Garcia and W.I. Zangwill,Pathways to Solutions, Fixed Points, and Equilibria (Prentice-Hall, Englewood Cliffs, NJ, 1981).Google Scholar
  26. [26]
    P. Hartman,Ordinary Differential Equations (Wiley, New York, 1964).Google Scholar
  27. [27]
    P. Hartman and G. Stampacchia, On some nonlinear elliptical differential functional equations, Acta Math. 115 (1966) 271–310.Google Scholar
  28. [28]
    A. Haurie and P. Marcotte, On the relationship between Nash-Cournot and Wardrop equilibria, Networks 15 (1985) 295–308.Google Scholar
  29. [29]
    D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Applications (Academic Press, New York, 1980).Google Scholar
  30. [30]
    H.J. Kushner and D.S. Clark,Stochastic Approximation Methods for Constrained and Unconstrained Systems (Springer, New York, 1978).Google Scholar
  31. [31]
    S. Lefschetz,Differential Equations. Geometric Theory (Interscience, New York, 1957).Google Scholar
  32. [32]
    A. Mas-Colell,The Theory of General Economic Equilibrium. A Differential Approach (Cambridge University Press, Cambridge, England, 1985).Google Scholar
  33. [33]
    A. Nagurney, Migration equilibrium and variational inequalities, Econ. Lett. 31 (1989) 109–112.Google Scholar
  34. [34]
    A. Nagurney,Network Economics: A Variational Inequality Approach (Kluwer Academic, Boston, MA, 1993).Google Scholar
  35. [35]
    A. Nagurney, J. Dong and M. Hughes, The formulation and computation of general financial equilibrium, Optimization 26 (1992) 339–354.Google Scholar
  36. [36]
    J.F. Nash, Equilibrium points inn-person games, Proc. National Acad. Sci. 36 (1950) 48–49.Google Scholar
  37. [37]
    J.F. Nash, Noncooperative games, Ann. Math. 54 (1951) 286–298.Google Scholar
  38. [38]
    K. Okuguchi,Expectations and Stability in Oligopoly Models, Lecture Notes in Economics and Mathematical Systems 138 (Springer, Berlin, 1976).Google Scholar
  39. [39]
    K. Okuguchi, The Cournot oligopoly and competitive equilibrium as solutions to non-linear complementarity problems, Econ. Lett. 12 (1983) 127–133.Google Scholar
  40. [40]
    K. Okuguchi and F. Szikdarovsky,The Theory of Oligopoly with Multi-Product Firms, Lecture Notes in Economics and Mathematical Systems 342 (Springer, Berlin, 1990).Google Scholar
  41. [41]
    P.A. Samuelson, The stability of equilibrium: Comparative statics and dynamics, Econometrica 9 (1941) 97–120.Google Scholar
  42. [42]
    A.V. Skorokhod, Stochastic equations for diffusions in a bounded region, Theory Prob. Appl. 6 (1961) 264–274.Google Scholar
  43. [43]
    S. Smale, A convergence process of price adjustment and global Newton methods, J. Math. Econ. 3 (1976) 107–120.Google Scholar
  44. [44]
    M.J. Smith, The existence and calculation of traffic equilibria, Transp. Res. 17B (1983) 291–303.Google Scholar
  45. [45]
    H.R. Varian, Dynamical systems with applications to economics, in:Handbook of Mathematical Economics 1, eds. K.J. Arrow and M.D. Intrilligator (North-Holland, Amsterdam, The Netherlands, 1981) pp. 93–110.Google Scholar
  46. [46]
    L.T. Watson, Solving the nonlinear complementarity problem by a homotopy method, SIAM J. Control Optim. 17 (1979) 36–46.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • Paul Dupuis
    • 1
  • Anna Nagurney
    • 2
  1. 1.Lefschetz Center for Dynamical Systems, Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.School of ManagementUniversity of MassachusettsAmherstUSA

Personalised recommendations