Facets of two-dimensional infinite group problems

Santanu S. Dey, Jean Philippe P. Richard

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


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 languageEnglish (US)
Pages (from-to)140-166
Number of pages27
JournalMathematics of Operations Research
Issue number1
StatePublished - Feb 2008
Externally publishedYes


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

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