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A class of Benders decomposition methods for variational inequalities

Author

Listed:
  • Juan Pablo Luna

    (COPPE-UFRJ, Engenharia de Produção)

  • Claudia Sagastizábal

    (IMECC - UNICAMP)

  • Mikhail Solodov

    (IMPA – Instituto de Matemática Pura e Aplicada)

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.

Suggested Citation

  • Juan Pablo Luna & Claudia Sagastizábal & Mikhail Solodov, 2020. "A class of Benders decomposition methods for variational inequalities," Computational Optimization and Applications, Springer, vol. 76(3), pages 935-959, July.
    • Handle: RePEc:spr:coopap:v:76:y:2020:i:3:d:10.1007_s10589-019-00157-y
    DOI: 10.1007/s10589-019-00157-y
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    References listed on IDEAS

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    1. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, December.
    2. Fuller, J. David & Chung, William, 2008. "Benders decomposition for a class of variational inequalities," European Journal of Operational Research, Elsevier, vol. 185(1), pages 76-91, February.
    3. William Chung & J. David Fuller, 2010. "Subproblem Approximation in Dantzig-Wolfe Decomposition of Variational Inequality Models with an Application to a Multicommodity Economic Equilibrium Model," Operations Research, INFORMS, vol. 58(5), pages 1318-1327, October.
    4. Regina S. Burachik & Alfredo N. Iusem, 2008. "Enlargements of Monotone Operators," Springer Optimization and Its Applications, in: Set-Valued Mappings and Enlargements of Monotone Operators, chapter 0, pages 161-220, Springer.
    5. Rahmaniani, Ragheb & Crainic, Teodor Gabriel & Gendreau, Michel & Rei, Walter, 2017. "The Benders decomposition algorithm: A literature review," European Journal of Operational Research, Elsevier, vol. 259(3), pages 801-817.
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    Cited by:

    1. Ernesto G. Birgin, 2020. "Preface of the special issue dedicated to the XII Brazilian workshop on continuous optimization," Computational Optimization and Applications, Springer, vol. 76(3), pages 615-619, July.

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