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.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Attouch, H., Théra, M.: A general duality principle for the sum of two operators. J. Convex Anal. 3(1), 1–24 (1996)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Auslender, A., Teboulle, M.: Lagrangian duality and related multiplier methods for variational inequality problems. SIAM J. Optim. 10(4), 1097–1115 (2000)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4(1), 238–252 (1962)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bonnans, J., Gilbert, J., Lemaréchal, C., Sagastizábal, C.: Numerical Optimization: Theoretical and Practical Aspects, Second edn. Springer, Berlin (2006)

    Google Scholar 

  5. 5.

    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)

    Google Scholar 

  6. 6.

    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)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    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)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Dantzig, G.B., Wolfe, P.: The decomposition algorithm for linear programs. Econom.: J. Econom. Soc. 29(4), 767–778 (1961)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Eckstein, J., Ferris, M.C.: Smooth methods of multipliers for complementarity problems. Math. Program. 86((1, Ser. A)), 65–90 (1999)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2007)

    Google Scholar 

  11. 11.

    Fuller, J.D., Chung, W.: Dantzig–Wolfe decomposition of variational inequalities. Comput. Econ. 25(4), 303–326 (2005)

    MATH  Google Scholar 

  12. 12.

    Fuller, J.D., Chung, W.: Benders decomposition for a class of variational inequalities. Eur. J. Oper. Res. 185(1), 76–91 (2008)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    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)

    MATH  Google Scholar 

  14. 14.

    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)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    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)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Mosco, U.: Dual variational inequalities. J. Math. Anal. Appl. 40, 202–206 (1972)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    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)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Sagastizábal, C.A., Solodov, M.V.: Parallel variable distribution for constrained optimization. Comput. Optim. Appl. 22(1), 111–131 (2002)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Solodov, M.V.: Constraint qualifications. In: Cochran, J.J., et al. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2010)

    Google Scholar 

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Mikhail Solodov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

Keywords

  • Variational inequalities
  • Benders decomposition
  • Dantzig–Wolfe decomposition
  • Stochastic Nash games

Mathematics Subject Classification

  • 90C33
  • 65K10
  • 49J53