A class of Benders decomposition methods for variational inequalities

Abstract

We develop new variants of Benders decomposition methods for variational inequality problems. The construction is done by applying the general class of Dantzig–Wolfe decomposition of Luna et al. (Math Program 143(1–2):177–209, 2014) to an appropriately defined dual of the given variational inequality, and then passing back to the primal space. As compared to previous decomposition techniques of the Benders kind for variational inequalities, the following improvements are obtained. Instead of rather specific single-valued monotone mappings, the framework includes a rather broad class of multi-valued maximally monotone ones, and single-valued nonmonotone. Subproblems’ solvability is guaranteed instead of assumed, and approximations of the subproblems’ mapping are allowed (which may lead, in particular, to further decomposition of subproblems, which may otherwise be not possible). In addition, with a certain suitably chosen approximation, variational inequality subproblems become simple bound-constrained optimization problems, thus easier to solve.