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Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization

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  • Egon Balas

Abstract

This is an overview of the significance and main uses of projection, lifting and extended formulation in integer and combinatorial optimization. Its first two sections deal with those basic properties of projection that make it such an effective and useful bridge between problem formulations in different spaces, i.e. different sets of variables. They discuss topics like projection and restriction, the integrality-preserving property of projection, the dimension of projected polyhedra, conditions for facets of a polyhedron to project into facets of its projections, and so on. The next two sections describe the use of projection for comparing the strength of different formulations of the same problem, and for proving the integrality of polyhedra by using extended formulations or lifting. Section 5 deals with disjunctive programming, or optimization over unions of polyhedra, whose most important incarnation are mixed 0-1 programs and their partial relaxations. It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as well as the sequential convexification of facial disjunctive programs, among them mixed 0-1 programs, with the related concept of disjunctive rank. Section 6 reviews lift-and-project cuts, the construction of cut generating linear programs, and techniques for lifting and for strengthening disjunctive cuts. Section 7 discusses the recently discovered possibility of solving the higher dimensional cut generating linear program without explicitly constructing it, by a sequence of properly chosen pivots in the simplex tableau of the linear programming relaxation. Finally, section 8 deals with different ways of combining cuts with branch and bound, and briefly discusses computational experience with lift-and-project cuts. Copyright Springer Science + Business Media, Inc. 2005

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  • Egon Balas, 2005. "Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization," Annals of Operations Research, Springer, vol. 140(1), pages 125-161, November.
  • Handle: RePEc:spr:annopr:v:140:y:2005:i:1:p:125-161:10.1007/s10479-005-3969-1
    DOI: 10.1007/s10479-005-3969-1
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    References listed on IDEAS

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    1. Balas, Egon & Jeroslow, Robert G., 1980. "Strengthening cuts for mixed integer programs," European Journal of Operational Research, Elsevier, vol. 4(4), pages 224-234, April.
    2. R.E. Bixby & W. Cook & A. Cox & E.K. Lee, 1999. "Computational experience with parallel mixed integerprogramming in a distributed environment," Annals of Operations Research, Springer, vol. 90(0), pages 19-43, January.
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    4. Adam N. Letchford, 2001. "On Disjunctive Cuts for Combinatorial Optimization," Journal of Combinatorial Optimization, Springer, vol. 5(3), pages 299-315, September.
    5. Williams, H. P., 1994. "An alternative explanation of disjunctive formulations," European Journal of Operational Research, Elsevier, vol. 72(1), pages 200-203, January.
    6. Egon Balas & Sebastián Ceria & Gérard Cornuéjols, 1996. "Mixed 0-1 Programming by Lift-and-Project in a Branch-and-Cut Framework," Management Science, INFORMS, vol. 42(9), pages 1229-1246, September.
    7. Beaumont, Nicholas, 1990. "An algorithm for disjunctive programs," European Journal of Operational Research, Elsevier, vol. 48(3), pages 362-371, October.
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    Cited by:

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    3. L. Foulds & B. Toklu & J. Wilson, 2009. "Modelling either-or relations in integer programming," Annals of Operations Research, Springer, vol. 166(1), pages 203-222, February.
    4. D. Aussel & J. J. Ye, 2008. "Quasiconvex Minimization on a Locally Finite Union of Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 1-16, October.
    5. STEPHAN, Rüdiger, 2010. "An extension of disjunctive programming and its impact for compact tree formulations," LIDAM Discussion Papers CORE 2010045, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Lee, Younsoo & Lee, Kyungsik, 2020. "Lot-sizing and scheduling in flat-panel display manufacturing process," Omega, Elsevier, vol. 93(C).
    7. Almeida, Maria Teresa & Carvalho, Filipa D., 2014. "An analytical comparison of the LP relaxations of integer models for the k-club problem," European Journal of Operational Research, Elsevier, vol. 232(3), pages 489-498.
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    10. Jan Kronqvist & Andreas Lundell & Tapio Westerlund, 2018. "Reformulations for utilizing separability when solving convex MINLP problems," Journal of Global Optimization, Springer, vol. 71(3), pages 571-592, July.

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