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Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

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Abstract

We present a generalization of the mixed integer rounding (MIR) approach for generating valid inequalities for (mixed) integer programming (MIP) problems. For any positive integer n, we develop n facets for a certain (n + 1)-dimensional single-constraint polyhedron in a sequential manner. We then show that for any n, the last of these facets (which we call the n-step MIR facet) can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, which we refer to as the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and günlük  (Math Program 105(1):29–53, 2006) are the first two families corresponding to n = 1,2, respectively. The n-step MIR inequalities are easily produced using periodic functions which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Finally, we prove that the n-step MIR inequalities generate two-slope facets for the infinite group polyhedra, and hence are potentially strong.

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References

  1. Aráoz, J., Evans, L., Gomory, R., Johnson, E.: Cyclic group and knapsack facets. Math. Programm. Ser. B 96, 377–408 (2003)

    Article  MATH  Google Scholar 

  2. Dash, S., günlük, O.: Valid inequalities based on simple mixed-integer sets. Math. Programm. 105(1), 29–53 (2006)

    Article  MATH  Google Scholar 

  3. Dash, S., günlük, O.: Valid inequalities based on the interpolation procedure. Math. Program. 106(1), 111–136 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Evans, L.: Cyclic group and knapsack facets with applications to cutting planes. Ph.D. thesis, Georgia Institute of Technology (2002)

  5. Gomory, R.E.: Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2, 451–558 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra I. Math. Program. 3, 23–85 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra II. Math. Program. 3, 359–389 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gomory, R.E., Johnson, E.L.: T-space and cutting planes. Math. Program. 96, 341–375 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gomory, R.E., Johnson, E.L., Evans, L.: Corner polyhedra and their connection with cutting planes. Math. Program. 96, 321–339 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Johnson, E.L.: On the group problem for mixed integer programming. Math. Program. Study 2, 137–179 (1974)

    Google Scholar 

  11. Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2001)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  13. Wolsey, L.A.: Integer Programming. Wiley, New York (1998)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX, 77843-3131, USA

    Kiavash Kianfar

  2. Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695-7906, USA

    Yahya Fathi

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  1. Kiavash Kianfar
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  2. Yahya Fathi
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Correspondence to Kiavash Kianfar.

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Kianfar, K., Fathi, Y. Generalized mixed integer rounding inequalities: facets for infinite group polyhedra. Math. Program. 120, 313–346 (2009). https://doi.org/10.1007/s10107-008-0216-y

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  • DOI: https://doi.org/10.1007/s10107-008-0216-y

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