Annals of Operations Research

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

Dynamical systems and variational inequalities

  • Paul Dupuis
  • Anna Nagurney
Methodological Advances

Abstract

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.

Keywords

Dynamical systems variational inequalities the Skorokhod Problem equilibrium problems 

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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

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