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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Attouch, H., Théra, M.: A general duality principle for the sum of two operators. J. Convex Anal. 3(1), 1–24 (1996)
Auslender, A., Teboulle, M.: Lagrangian duality and related multiplier methods for variational inequality problems. SIAM J. Optim. 10(4), 1097–1115 (2000)
Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4(1), 238–252 (1962)
Bonnans, J., Gilbert, J., Lemaréchal, C., Sagastizábal, C.: Numerical Optimization: Theoretical and Practical Aspects, Second edn. Springer, Berlin (2006)
Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators, Springer Optimization and Its Applications, vol. 8. Springer, New York (2008)
Çelebi, E., Fuller, J.D.: Master problem approximations in Dantzig–Wolfe decomposition of variational inequality problems with applications to two energy market models. Comput. Oper. Res. 40(11), 2724–2739 (2013)
Chung, W., Fuller, J.D.: Subproblem approximation in Dantzig–Wolfe decomposition of variational inequality models with an application to a multicommodity economic equilibrium model. Oper. Res. 58(5), 1318–1327 (2010)
Dantzig, G.B., Wolfe, P.: The decomposition algorithm for linear programs. Econom.: J. Econom. Soc. 29(4), 767–778 (1961)
Eckstein, J., Ferris, M.C.: Smooth methods of multipliers for complementarity problems. Math. Program. 86((1, Ser. A)), 65–90 (1999)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2007)
Fuller, J.D., Chung, W.: Dantzig–Wolfe decomposition of variational inequalities. Comput. Econ. 25(4), 303–326 (2005)
Fuller, J.D., Chung, W.: Benders decomposition for a class of variational inequalities. Eur. J. Oper. Res. 185(1), 76–91 (2008)
Gabriel, S.A., Fuller, J.D.: A Benders decomposition method for solving stochastic complementarity problems with an application in energy. Comput. Econ. 35(4), 301–329 (2010)
Luna, J.P., Sagastizábal, C., Solodov, M.: A class of Dantzig–Wolfe type decomposition methods for variational inequality problems. Math. Program. 143(1–2), 177–209 (2014)
Luna, J.P., Sagastizábal, C., Solodov, M.: An approximation scheme for a class of risk-averse stochastic equilibrium problems. Math. Program. 157(2), 451–481 (2016)
Mosco, U.: Dual variational inequalities. J. Math. Anal. Appl. 40, 202–206 (1972)
Rahmaniani, R., Crainic, T.G., Gendreau, M., Rei, W.: The Benders decomposition algorithm: a literature review. Eur. J. Oper. Res. 259(3), 801–817 (2017)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Sagastizábal, C.A., Solodov, M.V.: Parallel variable distribution for constrained optimization. Comput. Optim. Appl. 22(1), 111–131 (2002)
Solodov, M.V.: Constraint qualifications. In: Cochran, J.J., et al. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2010)
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.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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