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.
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The authors thank the two referees for their useful comments. All the three authors are supported by the FAPERJ Grant 203.052/2016 and by FAPERJ Grant E-26/210.908/2016 (PRONEX–Optimization). The second author’s research is also supported by CNPq Grant 303905/2015-8, by Gaspard Monge Visiting Professor Program, and by EPSRC Grant No. EP/ R014604/1 (Isaac Newton Institute for Mathematical Sciences, programme Mathematics for Energy Systems). The third author is also supported by CNPq Grant 303724/2015-3, and by Russian Foundation for Basic Research Grant 19-51-12003 NNIOa.
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Luna, J.P., Sagastizábal, C. & Solodov, M. A class of Benders decomposition methods for variational inequalities. Comput Optim Appl 76, 935–959 (2020). https://doi.org/10.1007/s10589-019-00157-y
- Variational inequalities
- Benders decomposition
- Dantzig–Wolfe decomposition
- Stochastic Nash games
Mathematics Subject Classification