A general method to apply Dantzig-Wolfe decomposition for variational inequalities with affine constraints

We develop a general method by which we can apply Dantzig-Wolfe (DW) decomposition method to any variational inequalities (VI). By adding dummy variables for creating linking affine constraints, the VI problem can be decomposed into a master equilibrium problem and a NLP subproblem. That is, the subproblem can be solved by any sophisticate NLP solvers. With making use of the Karush-Kuhn-Tucker (KKT) conditions of the decomposed problems in the Dantzig-Wolfe decomposition, we show that the simplicial decomposition for VI can be reduced to DW for the "dummy variable" version of VI. Consequently, both SD and DW might benefit by getting insights from each other. An illustrative example of applying both methods to asymmetric traffic assignment problems is provided.