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The Group-Theoretic Approach in Mixed Integer Programming

In: 50 Years of Integer Programming 1958-2008

Author

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  • Jean-Philippe P. Richard

    (University of Florida, Department of Industrial and Systems Engineering)

  • Santanu S. Dey

    (Université Catholique de Louvain, Center for Operations Research and Econometrics)

Abstract

In this chapter, we provide an overview of the mathematical foundations and recent theoretical and computational advances in the study of the grouptheoretic approach in mixed integer programming. We motivate the definition of group relaxation geometrically and present methods to optimize linear functions over this set. We then discuss fundamental results about the structure of group relaxations. We describe a variety of recent methods to derive valid inequalities for master group relaxations and review general proof techniques to show that candidate inequalities are strong (extreme) for these sets. We conclude by discussing the insights gained from computational studies aimed at gauging the strength of grouptheoretic relaxations and cutting planes for mixed integer programs.

Suggested Citation

  • Jean-Philippe P. Richard & Santanu S. Dey, 2010. "The Group-Theoretic Approach in Mixed Integer Programming," Springer Books, in: Michael Jünger & Thomas M. Liebling & Denis Naddef & George L. Nemhauser & William R. Pulleyblank & (ed.), 50 Years of Integer Programming 1958-2008, chapter 0, pages 727-801, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-68279-0_19
    DOI: 10.1007/978-3-540-68279-0_19
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