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An exact projection-based algorithm for bilevel mixed-integer problems with nonlinearities

Author

Listed:
  • Maximilian Merkert

    (Otto von Guericke University Magdeburg
    Technische Universität Braunschweig)

  • Galina Orlinskaya

    (Friedrich-Alexander University Erlangen-Nürnberg)

  • Dieter Weninger

    (Friedrich-Alexander University Erlangen-Nürnberg)

Abstract

We propose an exact global solution method for bilevel mixed-integer optimization problems with lower-level integer variables and including nonlinear terms such as, e.g., products of upper-level and lower-level variables. Problems of this type are extremely challenging as a single-level reformulation suitable for off-the-shelf solvers is not available in general. In order to solve these problems to global optimality, we enhance an approximative projection-based algorithm for mixed-integer linear bilevel programming problems from the literature to become exact under one additional assumption. This assumption still allows for discrete and continuous leader and follower variables on both levels, but forbids continuous upper-level variables to appear in lower-level constraints and thus ensures that a bilevel optimum is attained. In addition, we extend our exact algorithm to make it applicable to a wider problem class. This setting allows nonlinear constraints and objective functions on both levels under certain assumptions, but still requires that the lower-level problem is convex in its continuous variables. We also discuss computational experiments on modified library instances.

Suggested Citation

  • Maximilian Merkert & Galina Orlinskaya & Dieter Weninger, 2022. "An exact projection-based algorithm for bilevel mixed-integer problems with nonlinearities," Journal of Global Optimization, Springer, vol. 84(3), pages 607-650, November.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:3:d:10.1007_s10898-022-01172-w
    DOI: 10.1007/s10898-022-01172-w
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    References listed on IDEAS

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