Dantzig—Wolfe Decomposition of Variational Inequalities

Abstract

The creation and ongoing management of a large economic model can be greatly simplified if the model is managed in separate smaller pieces defined, e.g. by region or commodity. For this purpose, we define an extension of Dantzig--Wolfe decomposition for the variational inequality (VI) problem, a modeling framework that is widely used for models of competitive or oligopolistic markets. The subproblem, a collection of independent smaller models, is a relaxed VI missing some "difficult" constraints. The subproblem is modified at each iteration by information passed from the last solution of the master problem in a manner analogous to Dantzig--Wolfe decomposition for optimization models. The master problem is a VI which forms convex combinations of proposals from the subproblem, and enforces the difficult constraints. A valid stopping condition is derived in which a scalar quantity, called the "convergence gap," is monitored. The convergence gap is a generalization of the primal-dual gap that is commonly monitored in implementations of Dantzig--Wolfe decomposition for optimization models. Convergence is proved under conditions general enough to be applicable to many models. An illustration is provided for a two-region competitive model of Canadian energy markets.