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A recursive procedure to generate all cuts for 0–1 mixed integer programs

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Abstract

We study several ways of obtaining valid inequalities for mixed integer programs. We show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to Gomory's mixed integer cuts. We also show how all valid inequalities for mixed 0–1 programs can be generated recursively from a simple subclass of the disjunctive inequalities.

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References

  1. E. Balas, “Disjunctive programming: Cutting planes from logical conditions,” in: O. L. Mangasarian et al., eds.,Nonlinear Programming, Vol. 2 (Academic Press, New York, 1975) pp. 279–312.

    Google Scholar 

  2. C.E. Blair, “Two rules for deriving valid inequalities for 0–1 problems,”SIAM Journal of Applied Mathematics 31 (1976) 614–617.

    Google Scholar 

  3. V. Chvátal, “Edmonds polytopes and a hierarchy of combinatorial problems,”Discrete Mathematics 4 (1973) 305–337.

    Google Scholar 

  4. W. Cook, R. Kannan and A. Schrijver, “Chvátal closures for mixed integer programming problems,” Report No. 86444-OR, Institute for Econometrics and Operations Research, Bonn University (Bonn, 1987).

    Google Scholar 

  5. R. E. Gomory, “An algorithm for integer solutions to linear programs,” in: R. Graves and P. Wolfe, eds.,Recent Advances in Mathematical Programming (McGraw-Hill, New York, 1963) pp 269–302.

    Google Scholar 

  6. R.G. Jeroslow, “Cutting plane theory: Disjunctive methods,”Annals of Discrete Mathematics 1 (1972) 293–330.

    Google Scholar 

  7. R.G. Jeroslow, “Cutting plane theory: Algebraic methods,”Discrete Mathematics 23 (1978) 121–150.

    Google Scholar 

  8. E.L. Johnson, “On the group problem for mixed integer programming,”Mathematical Programming Study 2 (1974) 137–179.

    Google Scholar 

  9. G.L. Nemhauser and L.A. Wolsey, “A recursive procedure for generating all cuts for 0–1 mixed integer programs,” CORE DP 8439, Université Catholique de Louvain, (Louvain-la-Neuve, 1984).

    Google Scholar 

  10. A. Schrijver, “On cutting planes,”Annals of Discrete Mathematics 9 (1980) 291–296.

    Google Scholar 

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Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, 30332, Atlanta, GA, USA

    George L. Nemhauser

  2. Center for Operations Research and Econometrics, Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Laurence A. Wolsey

Authors
  1. George L. Nemhauser
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  2. Laurence A. Wolsey
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Additional information

The research of this author was supported by NSF Contract No. ECS-8540898.

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Nemhauser, G.L., Wolsey, L.A. A recursive procedure to generate all cuts for 0–1 mixed integer programs. Mathematical Programming 46, 379–390 (1990). https://doi.org/10.1007/BF01585752

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  • DOI: https://doi.org/10.1007/BF01585752

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