Chvátal closures for mixed integer programming problems

Abstract

Chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cutting-plane proofs is that for a polyhedron P and integral ve.:tor w, if max( wx Ix E P, wx integer}= I, then wx"' t is valid for all integral vectors in P. We consider the variant of this step where the requirement that wx be integer may be replaced by the requirement that wx be integer for some other integral vector w. The cutting-plane proofs thus obtained ma) be seen either as an abstraction of Gomory's mixed integer cutting-plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedron P, the set of vectors that satisfy every cutting plane for P with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chvatal closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.

Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

“…Split cuts [12], Gomory Mixed Integer (GMI) cuts [17], and Mixed Integer Rounding (MIR) cuts [23,24] are some of the most effective valid inequalities for Mixed Integer Linear Programming (MILP) [8]. While they are known to be equivalent [15,24], each of them provide different advantages and insights.…”

On the Practical Strength of Two-Row Tableau Cuts

“…One example of intersection cuts is the split cut [16]. A split cut is derived from the split set described in Proposition 2.1 by using (3).…”

“…This is because such instances can be decomposed row-wise and solved independently. Since each row has only one integer variable, the split closure is the convex hull; see [16].…”

A computational study of the cutting plane tree algorithm for general mixed-integer linear programs

“…Split cuts are obtained from disjunctions that are more general than the simple variable disjunctions, namely α x 1 ≤ β or α x 1 ≥ β +1, where (α, β) ∈ Z n 1 +1 and x 1 is the vector of integer variables. Despite the generality of the split disjunction, Cook et al (1990) provide an example to show that the split rank of a MILP-G could be infinite. Furthermore, the separation of split cuts is shown to be N P-hard (Caprara and Letchford, 2003).…”

“…This characterization is a generalization of the sequential convexification process of Balas (1979) for MILP with binary variables (MILP-B). However, it is important to note that the same sequential process of convexification (one variable at a time) does not yield the convex hull of MILP-G in finitely many steps (Owen and Mehrotra, 2001 Cook et al (1990) showed non-convergence for MILP-G using split disjunctions, and even in the case of MILP-B, Sen and Sherali (1985) presented examples of non-convergence in which facet inequalities of two-term (simple) disjunctions are derived to cut away the solution to the most recent LP relaxation. Chen et al (2009) use the CPT algorithm to solve the above instances in finitely many iterations.…”