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

摘要

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.