## Abstract

In this paper, we lay the foundation for the study of the two-dimensional mixed integer infinite group problem (2DMIIGP). We introduce tools to determine if a given continuous and piecewise linear function over the two-dimensional infinite group is subadditive and to determine whether it defines a facet of 2DMIIGP. We then present two different constructions that yield the first known families of facet-defining inequalities for 2DMIIGP. The first construction uses valid inequalities of the one-dimensional integer infinite group problem (1DIIGP) as building blocks for creating inequalities for the two-dimensional integer infinite group problem (2DIIGP). We prove that this construction yields all continuous piecewise linear facets of the two-dimensional group problem that have exactly two gradients. The second construction we present has three gradients and yields facet-defining inequalities of 2DMIIGP whose continuous coefficients are not dominated by those of facets of the one-dimensional mixed integer infinite group problem (1DMIIGP).

Original language | English (US) |
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Pages (from-to) | 140-166 |

Number of pages | 27 |

Journal | Mathematics of Operations Research |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2008 |

Externally published | Yes |

## Keywords

- Infinite group problem
- Mixed integer programs
- Multiple constraints
- Polyhedral theory