Balas introduced intersection cuts for mixed integer linear sets. Intersection cuts are given by closed form formulas and form an important class of cuts for solving mixed integer linear programs. In this paper we introduce an extension of intersection cuts to mixed integer conic quadratic sets. We identify the formula for the conic quadratic intersection cut by formulating a system of polynomial equations with additional variables that are satisfied by points on a certain piece of the boundary defined by the intersection cut. Using a software package from algebraic geometry we then eliminate variables from the system and get a formula for the intersection cut in dimension three. This formula is finally generalized and proved for any dimension. The intersection cut we present generalizes a conic quadratic cut introduced by Modaresi, Kilinc and Vielma.

Abstract Split cuts were introduced by Cook and his colleagues in the 1990s. Split cuts are based on the following simple Observation. Given an integer vector π ∈ ℤ n , and an integer π 0 ∈ ℤ, every integer point x ∈ ℤ n satisfies the split disjunction : either π T x ≤ π 0 or π T x ≥ π 0 + 1. Split cuts are a special case of the disjunctive cuts introduced by Balas . The set obtained by adding all split cuts to the linear program (LP) relaxation of a mixed integer program (MIP) is called the split closure . We show how split cuts can be obtained from a basis of the LP relaxation of a mixed integer program, and we describe the structure of the split closure. Furthermore, we give a summary of the computational results that have been obtained in the literature on the strength of the split closure as a relaxation of a MIP.

A maximal lattice free polyhedron L has max-facet-width equal to w if [Formula: see text] for all facets [Formula: see text] of L, and [Formula: see text] for some facet [Formula: see text] of L. The set obtained by adding all cuts whose validity follows from a maximal lattice free polyhedron with max-facet-width at most w is called the wth split closure. We show the wth split closure is a polyhedron. This generalizes a previous result showing this to be true when w = 1. We also consider the design of finite cutting plane proofs for the validity of an inequality. Given a measure of “size” of a maximal lattice free polyhedron, a natural question is how large a size s* of a maximal lattice free polyhedron is required to design a finite cutting plane proof for the validity of an inequality. We characterize s* based on the faces of the linear relaxation of the mixed integer linear set.

Many cuts used in practice to solve mixed integer programs are derived from a basis of the linear relaxation. Every such cut is of the form α T x ≥ 1, where x ≥ 0 is the vector of non-basic variables and α ≥ 0. For a point $\bar{x}$ of the linear relaxation, we call α T x ≥ 1 a zero-coefficient cut wrt. $\bar{x}$ if $\alpha^T \bar{x} = 0$ , since this implies α j = 0 when $\bar{x}_j > 0$ . We consider the following problem: Given a point $\bar{x}$ of the linear relaxation, find a basis, and a zero-coefficient cut wrt. $\bar{x}$ derived from this basis, or provide a certificate that shows no such cut exists. We show that this problem can be solved in polynomial time. We also test the performance of zero-coefficient cuts on a number of test problems. For several instances zero-coefficient cuts provide a substantial strengthening of the linear relaxation.

Providing a good formulation is an important part of solving a mixed-integer program.We suggest measuring the quality of a formulation by whether it is possible to strengthen the coefficients of the formulation. Sequentially strengthening coefficients can then be used as a tool for improving formulations.We believe this method could be useful for analyzing and producing tight formulations of problems that arise in practice.We illustrate the use of the approach on a problem in production scheduling. We also prove that coefficient strengthening leads to formulations with a desirable property: if no coefficient can be strengthened, then no constraint can be replaced by an inequality that dominates it. The effect of coefficient strengthening is tested on a number of problems in computational experiments. The strengthened formulations are compared to reformulations obtained by the preprocessor of a commercial software package. For several test problems, the formulations obtained by coefficient strengthening are substantially stronger than the formulations obtained by the preprocessor. In particular, we use coefficient strengthening to solve two difficult problems to optimality that have only recently been solved.

In Andersen et al. (Lecture Notes in Computer Science, vol. 4513, Springer, Berlin, pp. 1–15, 2007) we studied a mixed-integer set arising from two rows of a simplex tableau. We showed that facets of such a set can be obtained from lattice point free triangles and quadrilaterals associated with either three or four variables. In this paper we generalize our findings and show that, when upper bounds on the non-basic variables are also considered, further classes of facets arise that cannot be obtained from triangles and quadrilaterals. Specifically, when exactly one upper bound on a non-basic variable is introduced, stronger inequalities that can be derived from pentagons involving up to six variables also appear.

In this paper, we consider integral maximal lattice-free simplices. Such simplices have integer vertices and contain integer points in the relative interior of each of their facets, but no integer point is allowed in the full interior. In dimension three, we show that any integral maximal lattice-free simplex is equivalent to one of seven simplices up to unimodular transformation. For higher dimensions, we demonstrate that the set of integral maximal lattice-free simplices with vertices lying on the coordinate axes is finite. This gives rise to a conjecture that the total number of integral maximal lattice-free simplices is finite for any dimension.

In this paper we derive valid inequalities for general mixed-integer linear programs (MILPs) by considering several simplex tableau rows simultaneously. We explore links between these cutting planes and lattice-free convex sets which have a representation as the sum of a polytope and a linear space. The codimension of the linear space of such a set is called split-dimension. We show that in terms of a strength-measure introduced by Goemans [Math. Programming, 69 (1995), pp. 335-349], only lattice-free convex sets with full split-dimension give rise to a good approximation of the mixed-integer hull, whereas lattice-free convex sets with low split-dimension might approximate the mixed-integer hull arbitrarily badly, in general. However, we also show that ordinary split cuts approximate the mixed-integer hull to within a constant factor when the size of the input data is given.

We derive a certificate of integral infeasibility for linear systems with equations and inequalities by generating algebraically an outer description of a lattice point free polyhedron that contains the given integer infeasible system. The extension to the mixed integer setting is also derived.

A central result in the theory of integer optimization states that a system of linear diophantine equations Ax = b has no integral solution if and only if there exists a vector in the dual lattice, y T A integral such that y T b is fractional. We extend this result to systems that both have equations and inequalities {Ax = b, Cx d}. We show that a certificate of integral infeasibility is a linear system with rank(C) variables containing no integral point.

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.

Mixed-integer Gomory cuts have become an integral part of state-of-the-art software for solving mixed-integer linear programming problems. Therefore, improvements in the performance of these cutting planes can be of great practical value. In this paper, we present a simple and fast heuristic for improving the coefficients on the continuous variables in the mixed-integer Gomory cuts. This is motivated by the fact that in a mixed-integer Gomory cut, the coefficient of an integer variable lies between 0 and 1, whereas for a continuous variable, there is no upper bound. The heuristic tries to reduce the coefficients of the continuous variables. We call the resulting cuts reduce-and-split cuts. We found that on several test problems, reduce-and-split cuts can substantially enhance the performance of a branch-and-bound algorithm.

In the seventies, Balas introduced intersection cuts for a Mixed Integer Linear Program (MILP), and showed that these cuts can be obtained by a closed form formula from a basis of the standard linear programming relaxation. In the early nineties, Cook, Kannan and Schrijver introduced the split closure of an MILP, and showed that the split closure is a polyhedron. In this paper, we show that the split closure can be obtained using only intersection cuts. We give two different proofs of this result, one geometric and one algebraic. Furthermore, the result is used to provide a new proof of the fact that the split closure is a polyhedron. Finally, we extend the result to more general two-term disjunctions.