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Depth-Optimized Convexity Cuts

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  • Jonathan Eckstein
  • Mikhail Nediak

Abstract

This paper presents a general, self-contained treatment of convexity or intersection cuts. It describes two equivalent ways of generating a cut—via a convex set or a concave function—and a partial-order notion of cut strength. We then characterize the structure of the sets and functions that generate cuts that are strongest with respect to the partial order. Next, we specialize this analytical framework to the case of mixed-integer linear programming (MIP). For this case, we formulate two kinds of the deepest cut generation problem, via sets or via functions, and subsequently consider some special cases which are amenable to efficient computation. We conclude with computational tests of one of these procedures on a large set of MIPLIB problems. Copyright Springer Science + Business Media, Inc. 2005

Suggested Citation

  • Jonathan Eckstein & Mikhail Nediak, 2005. "Depth-Optimized Convexity Cuts," Annals of Operations Research, Springer, vol. 139(1), pages 95-129, October.
    • Handle: RePEc:spr:annopr:v:139:y:2005:i:1:p:95-129:10.1007/s10479-005-3445-y
    DOI: 10.1007/s10479-005-3445-y
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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.
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    3. 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.
    4. Philip B. Zwart, 1973. "Nonlinear Programming: Counterexamples to Two Global Optimization Algorithms," Operations Research, INFORMS, vol. 21(6), pages 1260-1266, December.
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