On the relative strength of families of intersection cuts arising from pairs of tableau constraints in mixed integer programs (original) (raw)

Cutting planes in integer and mixed integer programming

Discrete Applied Mathematics, 2002

This survey presents cutting planes that are useful or potentially useful in solving mixed integer programs. Valid inequalities for (i) general integer programs, (ii) problems with local structure such as knapsack constraints, and (iii) problems with 0-1 coe cient matrices, such as set packing, are examined in turn. Finally, the use of valid inequalities for classes of problems with structure, such as network design, is explored.

An optimality cut for mixed integer linear programs

European Journal of Operational Research, 1999

We derive the penalty cut, a simple optimality cut of general applicability in pure or mixed linear programs. This cut is tested on a number of examples and comparisons with the classical Gomory cut are provided.

Valid inequalities and reformulation techniques for Mixed Integer Nonlinear Programming

2015

One of the most important breakthroughs in the area of Mixed Integer Linear Programming (MILP) is the characterization of the convex hull of specially structured non-convex polyhedral sets in order to develop valid inequalities or cutting planes. Development of strong valid inequalities such as Split cuts, Gomory Mixed Integer (GMI) cuts, and Mixed Integer Rounding (MIR) cuts has resulted in highly effective branch-and-cut algorithms. While such cuts are known to be equivalent, each of their characterizations provides different advantages and insights. The study of cutting planes for Mixed Integer Nonlinear Programming (MINLP) is still much more limited than that for MILP, since characterizing cuts for MINLP requires the study of the convex hull of a non-convex and non-polyhedral set, which has proven to be significantly harder than the polyhedral case. However, there has been significant work on the computational use of cuts in MINLP. Furthermore, there has recently been a signific...

Conjunctive Cuts for Integer Programs

This paper deals with a family of conjunctive inequalities. Such inequalities are needed to describe the polyhedron associated with all the integer points that satisfy several knapsack constraints simultaneously. Here we demonstrate the strength and potential of conjunctive inequalities in connection with lifting from a computational point of view. 1

Foundation-penalty cuts for mixed-integer programs

Operations Research Letters, 2003

We propose a new class of Foundation-Penalty (FP) cuts for GUB-constrained (and ordinary) mixed-integer programs, which are easy to generate by exploiting standard penalty calculations that are routinely employed in branch-and-bound contexts. The FP cuts are derived with reference to a selected integer variable or GUB set, and a foundation function that is typically a reduced cost function corresponding to an optimal linear programming basis. The concept behind the generation of these cuts generalizes the lifting process, and as we demonstrate, bears relationships with other classical cuts such as disjunctive cuts, lift-and-project cuts, convexity cuts, Gomory cuts, and mixedinteger rounding cuts. For example, the easily derived but often useful cutting planes at the level of Gomory cuts and mixed integer rounding cuts are subsumed by related FP cuts that simply 'plug in' penalty values from standard calculations (where the penalties are allowed to go beyond those from the first primitive mixed-integer programming codes). In general, the strength of these FP cuts can be varied according to the trade-offs between the strengths of alternative penalty calculations and the effort required to compute them, by virtue of the interactions between the foundation function and the branching disjunctions that underlie the penalty computations. By this means, FP cuts are especially convenient for use in branch-and-cut procedures, where penalty calculations are employed as a matter of course, and afford new strategies for generating cutting planes in this setting.

Projected Chvátal-Gomory cuts for mixed integer linear programs

Mathematical Programming, 2008

Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure integer linear program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the Chvátal–Gomory (CG) separation problem as a mixed integer linear program (MILP) which is then solved by a general- purpose MILP solver. Unfortunately, this approach does not extend immediately to the Gomory mixed integer (GMI) closure of an MILP, since the GMI separation problem involves the solution of a nonlinear mixed integer program or a parametric MILP. In this paper we introduce a projected version of the CG cuts, and study their practical effectiveness for MILP problems. The idea is to project first the linear programming relaxation of the MILP at hand onto the space of the integer variables, and then to derive Chvátal–Gomory cuts for the projected polyhedron. Though theoretically dominated by GMI cuts, projected CG cuts have the advantage of producing a separation model very similar to the one introduced by Fischetti and Lodi, which can typically be solved in a reasonable amount of computing time.

Penalty Cuts for GUB-Constrained Mixed Integer Programs

Penalty cuts provide a new class of cutting planes for GUB-constrained (and ordinary) mixed integer programs, which are easy to generate by exploiting standard penalty calculations employed in branch-and-bound. The Penalty cuts are created by reference to a selected GUB set and a foundation hyperplane that is typically dual feasible relative to a current linear programming basis. As a special case, the GUB restrictions translate into related disjunctions that provide cutting planes for ordinary MIP problems. At the simplest level these yield the classical Gomory mixed-integer cuts, and at higher levels yield deeper cuts. In general, the strength of the cuts can be varied according to the tradeoffs between the strengths of alternative penalty calculations and the effort required to apply them, according to interactions between the foundation hyperplanes and the branching disjunctions that underlie the penalties. By this means, Penalty cuts are especially convenient to use in branch-and-cut procedures, where penalty calculations are employed as a matter of course, and afford new strategies for generating cutting planes in this setting.

Local cuts for mixed-integer programming

Mathematical Programming Computation, 2013

A general framework for cutting-plane generation was proposed by Applegate et al. in the context of the traveling salesman problem. The process considers the image of a problem space under a linear mapping, chosen so that a relaxation of the mapped problem can be solved efficiently. Optimization in the mapped space can be used to find a separating hyperplane if one exists, and via substitution this gives a cutting plane in the original space. We apply this procedure to general mixed-integer programming problems, obtaining a range of possibilities for new sources of cutting planes.

A recursive procedure to generate all cuts for 0���1 mixed integer programs

1990

We study several ways of obtaining valid inequalities for mixed integer programs. We show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to Gomory's mixed integer cuts. We also show how all valid inequalities for mixed 0 1 programs can be generated recursively from a simple subclass of the disjunctive inequalities.

Finite Disjunctive Programming Characterizations for General Mixed-Integer Linear Programs

Operations Research, 2011

In this paper, we give a finite disjunctive programming procedure to obtain the convex hull of general mixed-integer linear programs (MILP) with bounded integer variables. We propose a finitely convergent convex hull tree algorithm which constructs a linear program that has the same optimal solution as the associated MILP. In addition, we combine the standard notion of sequential cutting planes with ideas underlying the convex hull tree algorithm to help guide the choice of disjunctions to use within a cutting plane method. This algorithm, which we refer to as the cutting plane tree algorithm, is shown to converge to an integral optimal solution in finitely many iterations. Finally, we illustrate the proposed algorithm on three well-known examples in the literature that require an infinite number of elementary or split disjunctions in a rudimentary cutting plane algorithm.