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A Unified Framework for Multistage Mixed Integer Linear Optimization

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Bilevel Optimization

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 161))

Abstract

We introduce a unified framework for the study of multilevel mixed integer linear optimization problems and multistage stochastic mixed integer linear optimization problems with recourse. The framework highlights the common mathematical structure of the two problems and allows for the development of a common algorithmic framework. Focusing on the two-stage case, we investigate, in particular, the nature of the value function of the second-stage problem, highlighting its connection to dual functions and the theory of duality for mixed integer linear optimization problems, and summarize different reformulations. We then present two main solution techniques, one based on a Benders-like decomposition to approximate either the risk function or the value function, and the other one based on cutting plane generation.

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Notes

  1. 1.

    When the value function is not real-valued everywhere, we have to show that there is a real-valued function that coincides with the value function when it is real-valued and is itself real-valued everywhere else, but is still a feasible dual function (see [140]).

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Authors and Affiliations

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

    Suresh Bolusani & Ted K. Ralphs

  2. Department of Mathematical Sciences, University of Southampton, Southampton, UK

    Stefano Coniglio

  3. Lindo Systems, Inc., Chicago, IL, USA

    Sahar Tahernejad

Authors
  1. Suresh Bolusani
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  2. Stefano Coniglio
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  3. Ted K. Ralphs
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  4. Sahar Tahernejad
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Correspondence to Stefano Coniglio .

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  1. Institute of Numerical Mathematics and Optimization, TU Bergakademie Freiberg, Freiberg, Germany

    Stephan Dempe

  2. School of Mathematical Sciences, University of Southampton, Southampton, UK

    Alain Zemkoho

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Bolusani, S., Coniglio, S., Ralphs, T.K., Tahernejad, S. (2020). A Unified Framework for Multistage Mixed Integer Linear Optimization. In: Dempe, S., Zemkoho, A. (eds) Bilevel Optimization. Springer Optimization and Its Applications, vol 161. Springer, Cham. https://doi.org/10.1007/978-3-030-52119-6_18

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