Approximate Dantzig–Wolfe decomposition to solve a class of variational inequality problems with an illustrative application to electricity market models

Abstract

In this study, we develop a new Approximate Dantzig–Wolfe (ADW) decomposition method for variational inequalities (VI) based on the work by Chung and Fuller (2010) and Çelebi and Fuller (2013). The decomposed VI consists of one approximate subproblem, which is nonlinear programming (NLP) or linear programming (LP) and one approximate master problem, which is an NLP. Note that we can have many approximate subproblems depending upon the approximation method and the structure of the constraint set. On the other hand, if the VI mapping in the approximate master problem is equal to that in the iterative methods for solving VI, then the ADW-VI simply consists of the computational sequence of solving NLP (or LP) subproblem(s) and NLP master problem in an iterative manner. That is, the iterative methods for VI and the DW decomposition method are combined into a single iterative loop. The details of the method are presented as well as an extension of the theory from Chung and Fuller (2010) and Çelebi and Fuller (2013). In addition, numerical results are provided based on two time-of-use pricing models of Ontario electricity market in Çelebi and Fuller (2013), but for which the new master problem approximation different from Çelebi and Fuller (2013) has been used. These results validate ADW-VI, and in some computational instances, indicates dramatic improvements in solution times as compared to reference methods, like diagonalization method of Dafermos (1983). Another set of numerical results, based on a simple electricity market, illustrates that ADW-VI can be faster than the PATH solver when solving large-scale problem instances.