On degenerate multi-row Gomory cuts

Borozan and Cornuéjols introduced a semi-infinite relaxation for mixed-integer linear programs, derived from the simplex tableau relative to a fractional vertex f of the linear relaxation. They showed that any minimal valid inequality for such relaxation is determined by a maximal lattice-free convex set containing f . In this note we show that, for the purpose of generating cuts for the original MIP, one needs to consider only maximal lattice-free convex bodies with f in the interior. Most cutting plane algorithms use cuts, such as Gomory’s Mixed Integer Cuts, deduced by one single tableau inequality by arguments on the integrality and nonnegativity of some of the variables involved. The idea of studying inequalities that can be inferred by multiple rows of the simplex tableau has recently gained interest through the work of Dey and Richard [4], Dey et Al. [5], Andersen et Al. [1], Borozan and Cornuéjols [2], Cornuéjols and Margot [3], Dey and Wolsey [6], while Espinoza [7] reports promising computational results. The following model, obtained from the simplex tableau by dropping the nonnegativity constraints on the basic variables and the integrality constraints on the non-basic ones, has been studied by Andersen et Al. [1] in the case of two tableau rows, and their results have been generalized by Borozan and Cornuéjols [2] to the case of multiple rows; x = f + ∑k j=1 r sj, x ∈ Z, s ∈ R+. (1) Here f, r, . . . , r ∈ Q and f / ∈ Z. Let Rf (r, . . . , r) be the convex hull of vectors s ∈ R for which there exists x ∈ R such that (x, s) satisfies (1). Since all data are rational, ∗Dipartimento di Matematica Pura e Applicata, Universitá di Padova, Via Trieste 63, 35121 Padova, Italy. ([email protected])