Inequalities from Two Rows of a Simplex Tableau

Abstract

Abstract. In this paper we explore the geometry of the integer points in a cone rooted at a rational point. This basic geometric object allows us to establish some links between lattice point free bodies and the derivation of inequalities for mixed integer linear programs by considering two rows of a simplex tableau simultaneously.

A Geometric Perspective on Lifting

“…is valid for (1) if it holds for every (x, s, y) satisfying (1). If (2) is valid, we say that the function (ψ, π) is valid for (1).…”

“…If (2) is valid, we say that the function (ψ, π) is valid for (1). A valid function (ψ, π) is minimal if there is no valid function (ψ , π ) distinct from (ψ, π) such that ψ (r) ≤ ψ(r), π (r) ≤ π(r) for all r ∈ R n .…”

“…Information about valid inequalities for (1) automatically transfers to the problem of cutting-off a fractional basic solution of the linear programming relaxation. Most cutting planes used in practice (Gomory mixed integer cuts, Mixed Integer Rounding inequalities, knapsack covers, flow covers, lift-and-project cuts and many other) are valid for Gomory's corner polyhedron, which is the convex hull of solutions to (1) where S = Z n and all but a finite number of the variables s r and y r are set to 0.…”

Minimal Valid Inequalities for Integer Constraints

“…Most general purpose cutting planes used in state-of-the-art solvers are obtained by generating a linear combination of the original constraints Ax = b, and by applying integrality arguments to the resulting equation (Gomory's Mixed Integer Cuts, MIR inequalities and split cuts are examples). Recently, there has been interest in cutting planes that cannot be deduced from a single equation, but can be deduced by integrality arguments involving two equations (Dey and Richard [5], Andersen, Louveaux, Weismantel and Wolsey [1]). Let us consider the following relaxation of the integer program (IP) starting from its equivalent form (1).…”

“…Recently, there has been interest in cutting planes that cannot be deduced from a single equation, but can be deduced by integrality arguments involving two equations (Dey and Richard [5], Andersen, Louveaux, Weismantel and Wolsey [1]). Let us consider the following relaxation of the integer program (IP) starting from its equivalent form (1). We drop the nonnegativity restriction on all basic variables x i , i ∈ B, and we drop the integrality restriction on all the nonbasic variables x j , j ∈ J.…”