Differential Variational Inequalities and Differential Nash Games

  • Terry L. Friesz
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 135)


In this chapter we focus on extending the notion of a noncooperative Nash equilibrium to a dynamic, continuous-time setting. The dominant mathematical perspective we will employ is that of a differential variational inequality. In fact we shall see that many of the results obtained in the previous chapter for finite-dimensional variational inequalities and static games carry over with some slight modifications to the dynamic, continuous-time setting we now address.


Variational Inequality Optimal Control Problem Variational Inequality Problem Nonlinear Complementarity Problem Mixed Variational Inequality 
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.

List of References Cited and Additional Reading

  1. Adams, R. A. (1970). Equivalent norms for Sobolev spaces. Proceedings of the American Mathematical Society 24(1), 63–66.Google Scholar
  2. Aubin, J. P. and A. Cellina (1984). Differential Inclusions. New York: Springer-Verlag.Google Scholar
  3. Bernstein, D., T. L. Friesz, R. L. Tobin, and B. W. Wie (1993). A variational control formulation of the simultaneous route and departure-time equilibrium problem. Proceedings of the International Symposium on Transportation and Traffic Theory, 107–126.Google Scholar
  4. Bressan, A. and B. Piccoli (2007). An Introduction to the Mathematical Theory of Control. Springfield, MO: American Institute of Mathematical Sciences.Google Scholar
  5. Brouwer, L. E. J. (1910). Ueber eineindeutige, stetige transformationen von flächen in sich. Mathematische Annalen 69(2), 176–180.CrossRefGoogle Scholar
  6. Browder, F. E. (1968). The fixed point theory of multivalued mappings in topological vector spaces. Mathematische Annalen 177, 283–301.CrossRefGoogle Scholar
  7. Friesz, T., D. Bernstein, and R. Stough (1996). Dynamic systems, variational inequalities, and control-theoretic models for predicting time-varying urban network flows. Transportation Science 30(1), 14–31.CrossRefGoogle Scholar
  8. Friesz, T. and R. Mookherjee (2006). Differential variational inequalities with state-dependent time shifts. Transportation Research Part B.Google Scholar
  9. Friesz, T. L., D. Bernstein, T. Smith, R. Tobin, and B. Wie (1993). A variational inequality formulation of the dynamic network user equilibrium problem. Operations Research 41(1), 80–91.CrossRefGoogle Scholar
  10. Fukushima, M. (1992). Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming 53, 99–110.CrossRefGoogle Scholar
  11. Isaacs, R. (1965). Differential Games. New York: Dover.Google Scholar
  12. Kachani, S. and G. Perakis (2002a). A fluid model of dynamic pricing and inventory management for make to stock manufacturing systems. Technical report, Sloan School of Management, MIT.Google Scholar
  13. Kachani, S. and G. Perakis (2002b). Fluid dynamics models and their application in transportation and pricing. Technical report, Sloan School of Management, MIT.Google Scholar
  14. Konnov, I. V. and S. Kum (2001). Descent methods for mixed variational inequalities in a Hilbert space. Nonlinear Analysis: Theory, Methods and Applications 47(1), 561–572.CrossRefGoogle Scholar
  15. Konnov, I. V., S. Kum, and G. M. Lee (2002). On convergence of descent methods for variational inequalities in a Hilbert space. Mathematical Methods of Operations Research 55, 371–382.CrossRefGoogle Scholar
  16. Kwon, C., T. Friesz, R. Mookherjee, T. Yao, and B. Feng (2009). Non-cooperative competition among revenue maximizing service providers with demand learning. European Journal of Operational Research 197(3), 981–996.CrossRefGoogle Scholar
  17. Minoux, M. (1986). Mathematical Programming: Theory and Algorithms. New York: John Wiley.Google Scholar
  18. Mookherjee, R. and T. Friesz (2008). Pricing, allocation, and overbooking in dynamic service network competition when demand is uncertain. Production and Operations Management 14(4), 1–20.Google Scholar
  19. Patriksson, M. (1997). Merit functions and descent algorithms for a class of variational inequality problems. Optimization 41, 37–55.CrossRefGoogle Scholar
  20. Peng, J.-M. (1997). Equivalence of variational inequality problems to unconstrained minimization. Mathematical Programming 78, 347–355.Google Scholar
  21. Perakis, G. (2000). The dynamic user equilibrium problem through hydrodynamic theory. Sloan School of Management, MIT, preprint.Google Scholar
  22. Pshenichnyi, B. N. (1971). Necessary Conditions for an Extremum. New York: Marcel Dekker.Google Scholar
  23. Todd, M. J. (1976). The Computation of Fixed Points and Applications. New York: Springer-Verlag.Google Scholar
  24. Yamashita, N., K. Taji, and M. Fukushima (1997). Unconstrained optimization reformulations of variational inequality problems. Journal of Optimization Theory and Applications 92(3), 439–456.CrossRefGoogle Scholar
  25. Zhu, D. L. and P. Marcotte (1994). An extended descent framework for variational inequalities. Journal of Optimization Theory and Applications 80(2), 349–366.CrossRefGoogle Scholar
  26. Zhu, D. L. and P. Marcotte (1998). Convergence properties of feasible descent methods for solving variational inequalities in Banach spaces. Computational Optimization and Applications 10(1), 35–49.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. Industrial & Manufacturing EngineeringPennsylvania State UniversityUniversity ParkUSA

Personalised recommendations