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Mixed-Integer Cuts from Cyclic Groups

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Book cover Integer Programming and Combinatorial Optimization (IPCO 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3509))

Abstract

We analyze a separation procedure for Mixed-Integer Programs related to the work of Gomory and Johnson on interpolated subadditive functions. This approach has its roots in the Gomory-Johnson characterization on the master cyclic group polyhedron. To our knowledge, the practical benefit that can be obtained by embedding interpolated subadditive cuts in a cutting plane algorithm was not investigated computationally by previous authors. In this paper we compute, for the first time, the lower bound value obtained when adding (implicitly) all the interpolated subadditive cuts that can be derived from the individual rows of an optimal LP tableau, thus approximating the optimization over the so-called Gomory’s Corner polyhedron. The computed bound is compared with that obtained when only Gomory mixed-integer cuts are used, on a large test-bed of MIPLIB instances.

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References

  1. Andreello, G., Caprara, A., Fischetti, M.: Embedding Cuts in a Branch and Cut Framework: a Computational Study with {0,1/2}-Cuts, Technical Report, University of Padova, available at (2003), http://www.dei.unipd.it/~fisch/papers

  2. Araoz, J., Gomory, R.E., Johnson, E.L., Evans, L.: Cyclic group and knapsack facets. Mathematical Programming 96, 377–408 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Balas, E., Ceria, S., Cornuejols, G., Natraj, N.: Gomory Cuts Revisited. Operations Research Letters 19, 1–9 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cornuejols, G., Li, Y., Vandenbussche, D.: K-Cuts: A Variation of Gomory Mixed Integer Cuts from the LP Tableau. INFORMS Journal on Computing 15, 385–396 (2003)

    Article  MathSciNet  Google Scholar 

  5. Dash, S., Gunluk, O.: Valid inequalities based on simple mixed-integer sets, IBM Research Report RC22922, T. J. Watson Research Center, Yorktown Heights, New York (2003)

    Google Scholar 

  6. Dash, S., Gunluk, O.: Valid inequalities based on the interpolation procedure, IBM Research Report RC22922, T. J. Watson Research Center, Yorktown Heights, New York (2004)

    Google Scholar 

  7. Evans, L.: Cyclic Groups and Knapsack Facets with Applications to Cutting Planes, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, Georgia (2002)

    Google Scholar 

  8. Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bulletin of the AMS 64, 275–278 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gomory, R.E.: An algorithm for the mixed-integer problem. Report RM-2597, Rand Corporation (1960)

    Google Scholar 

  10. Gomory, R.E.: Some Polyhedra Related to Combinatorial Problems. Journal of Linear Algebra and its Applications 2, 451–558 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gomory, R.E., Johnson, E.L.: Some Continuous Functions Related to Corner Polyhedra I. Mathematical Programming 3, 23–85 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gomory, R.E., Johnson, E.L.: Some Continuous Functions Related to Corner Polyhedra II. Mathematical Programming 3, 359–389 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gomory, R.E., Johnson, E.L.: T-space and cutting planes. Mathematical Programming 96, 341–375 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gomory, R.E., Johnson, E., Evans, L.: Corner Polyhedra and their connection with cutting planes. Mathematical Programming 96, 321–339 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. ILOG Cplex 9.0: User’s Manual and Reference Manual, ILOG, S.A. (2004), http://www.ilog.com/

  16. ILOG Concert Technology 2.0: User’s Manual and Reference Manual, ILOG, S.A. (2004), http://www.ilog.com/

  17. Letchford, A.N., Lodi, A.: Strengthening Chvàtal-Gomory cuts and Gomory fractional cuts. Operations Research Letters 30(2), 74–82 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Koppe, M., Louveaux, Q., Weismantel, R., Wolsey, L.A.: Extended Formulations for Gomory Corner Polyhedra. Discrete Optimization Discrete Optimization 1, 141–165 (2004)

    MathSciNet  Google Scholar 

  19. MIPLIB - Mixed Integer Problem Library 2003 (2003), http://miplib.zib.de

  20. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. DEI, Department of Information Engineering, University of Padua, via Gradenigo 6/a, I-35100, Padova, Italy

    Matteo Fischetti & Cristiano Saturni

Authors
  1. Matteo Fischetti
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  2. Cristiano Saturni
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Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität zu Köln,  

    Michael Jünger

  2. Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Volker Kaibel

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© 2005 Springer-Verlag Berlin Heidelberg

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Fischetti, M., Saturni, C. (2005). Mixed-Integer Cuts from Cyclic Groups. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_1

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  • DOI: https://doi.org/10.1007/11496915_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26199-5

  • Online ISBN: 978-3-540-32102-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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