Approximation Methods in Dantzig—Wolfe Decomposition of Variational Inequalities—A Review and Extension

In this study, we review some approximation methods being used in Dantzig-Wolfe (DW) decomposition method for variational inequalities (VI). After applying DW decomposition method, the decomposed VI consists of one VI subproblem (sub-VI) and one VI master problem (master-VI). In each decomposition computational loop, we need to use an iterative method to solve both sub-VI and master-VI individually. To improve the computational efficiency, approximation methods in solving sub-VI or master-VI (not both) are used from the literature. Under the approximation methods, the approximate sub-VI is a LP or NLP. On the other hand, master-VI is approximately solved until a condition being met. Since both approximation methods for sub-VI and master-VI were developed separately, there is a knowledge gap that if both approximation methods can be applied at the same time in solving VI with DW decomposition method. The current study is to fill this gap. That is, we propose to apply both approximation methods of sub-VI and master-VI in one DW decomposition loop. An illustrative application is provided.