Variational Inequality Problems

The purpose of this chapter is to explain variational inequality (VI) formulations of equilibrium problems, and the close connection of a VI problem to an equivalent complementarity problem. There are sometimes advantages to a VI formulation compared to a complementarity formulation: the complementarity formulation has primal decision variables, and dual variables that arise, e.g., when specifying the KKT conditions of individual agents; but a VI formulation has the same primal variables, with few, or no dual variables, which can considerably ease the coding of the model in GAMS. This coding advantage is particularly evident when implementing large complex models, or decomposition algorithms as discussed in Chapter 9. However, the derivation of a complementarity model formulation is usually easier than the derivation of an equivalent VI model: e.g., many complementarity models in this book are derived by writing down the KKT conditions of the agents, together with market-clearing conditions, but the procedure to write down a VI model with few or no dual variables is not as easily stated. In this chapter, we alleviate this difficulty by showing how to arrive quickly at the formulation of a VI model for a large class of Nash equilibrium and generalized Nash equilibrium settings.