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A framework for generalized Benders’ decomposition and its application to multilevel optimization

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Abstract

We describe a framework for reformulating and solving optimization problems that generalizes the well-known framework originally introduced by Benders. We discuss details of the application of the procedures to several classes of optimization problems that fall under the umbrella of multilevel/multistage mixed integer linear optimization problems. The application of this abstract framework to this broad class of problems provides new insights and a broader interpretation of the core ideas, especially as they relate to duality and the value functions of optimization problems that arise in this context.

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Notes

  1. We use the term “stage” in describing the decision epochs of an MMILP, rather than the alternative “level” used in the multilevel optimization literature because of its broader connotation and connection to stochastic optimization.

  2. When the value function is not real-valued everywhere, we have to show that there exists a real-valued function that coincides with the value function whenever the value function is real-valued and is itself real-valued everywhere else, but is still a feasible dual function (see Wolsey [64]).

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Acknowledgements

This research was made possible with support from National Science Foundation Grants CMMI-1435453, CMMI-0728011, and ACI-0102687, as well as Office of Naval Research Grant N000141912330.

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  1. Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA

    Suresh Bolusani & Ted K. Ralphs

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Bolusani, S., Ralphs, T.K. A framework for generalized Benders’ decomposition and its application to multilevel optimization. Math. Program. 196, 389–426 (2022). https://doi.org/10.1007/s10107-021-01763-7

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