Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

  • Authors:

  • Kiavash Kianfar;Yahya Fathi

  • Affiliations:

  • Texas A&M University, Department of Industrial and Systems Engineering, 77843-3131, College Station, TX, USA;North Carolina State University, Department of Industrial and Systems Engineering, 27695-7906, Raleigh, NC, USA

  • Venue:
  • Mathematical Programming: Series A and B
  • Year:
  • 2009

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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.