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A Polyhedral Study of Binary Polynomial Programs

Author

Listed:
  • Alberto Del Pia

    (Department of Industrial and Systems Engineering and Wisconsin Institute for Discovery, University of Wisconsin–Madison, Madison, Wisconsin 53706)

  • Aida Khajavirad

    (Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213)

Abstract

We study the polyhedral convex hull of a mixed-integer set 𝒮 defined by a collection of multilinear equations over the unit hypercube. Such sets appear frequently in the factorable reformulation of mixed-integer nonlinear optimization problems. In particular, the set 𝒮 represents the feasible region of a linearized unconstrained binary polynomial optimization problem. We define an equivalent hypergraph representation of the mixed-integer set 𝒮 , which enables us to derive several families of facet-defining inequalities, structural properties, and lifting operations for its convex hull in the space of the original variables. Our theoretical developments extend several well-known results from the Boolean quadric polytope and the cut polytope literature, paving a way for devising novel optimization algorithms for nonconvex problems containing multilinear sub-expressions.

Suggested Citation

  • Alberto Del Pia & Aida Khajavirad, 2017. "A Polyhedral Study of Binary Polynomial Programs," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 389-410, May.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:2:p:389-410
    DOI: 10.1287/moor.2016.0804
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    References listed on IDEAS

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    1. Endre Boros & Peter L. Hammer, 1993. "Cut-Polytopes, Boolean Quadric Polytopes and Nonnegative Quadratic Pseudo-Boolean Functions," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 245-253, February.
    2. Faiz A. Al-Khayyal & James E. Falk, 1983. "Jointly Constrained Biconvex Programming," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 273-286, May.
    3. Sonia Cafieri & Jon Lee & Leo Liberti, 2010. "On convex relaxations of quadrilinear terms," Journal of Global Optimization, Springer, vol. 47(4), pages 661-685, August.
    4. Keith Zorn & Nikolaos Sahinidis, 2014. "Global optimization of general nonconvex problems with intermediate polynomial substructures," Journal of Global Optimization, Springer, vol. 59(2), pages 673-693, July.
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    Cited by:

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    2. Leonardo Lozano & David Bergman & J. Cole Smith, 2020. "On the Consistent Path Problem," Operations Research, INFORMS, vol. 68(6), pages 1913-1931, November.

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