Intersection Cuts Research Articles

On the implementation and strengthening of intersection cuts for QCQPs

On the implementation and strengthening of intersection cuts for QCQPs

Intersection Disjunctions for Reverse Convex Sets

We present a framework to obtain valid inequalities for a reverse convex set: the set of points in a polyhedron that lie outside a given open convex set. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. An intersection cut is a well-known valid inequality for a reverse convex set that is generated from a basic solution that lies within the convex set. We introduce a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We first propose an extension to intersection cuts that defines a two-term disjunction for a reverse convex set, which we refer to as an intersection disjunction. Next, we generalize this analysis to a multiterm disjunction by considering the convex set’s recession directions. These disjunctions can be used in a cut-generating linear program to obtain valid inequalities for the reverse convex set.

Multirow Intersection Cuts Based on the Infinity Norm

When generating multirow intersection cuts for mixed-integer linear optimization problems, an important practical question is deciding which intersection cuts to use. Even when restricted to cuts that are facet defining for the corner relaxation, the number of potential candidates is still very large, especially for instances of large size. In this paper, we introduce a subset of intersection cuts based on the infinity norm that is very small, works for relaxations having arbitrary number of rows and, unlike many subclasses studied in the literature, takes into account the entire data from the simplex tableau. We describe an algorithm for generating these inequalities and run extensive computational experiments in order to evaluate their practical effectiveness in real-world instances. We conclude that this subset of inequalities yields, in terms of gap closure, around 50% of the benefits of using all valid inequalities for the corner relaxation simultaneously, but at a small fraction of the computational cost, and with a very small number of cuts. Summary of Contribution: Cutting planes are one of the most important techniques used by modern mixed-integer linear programming solvers when solving a variety of challenging operations research problems. The paper advances the state of the art on general-purpose multirow intersection cuts by proposing a practical and computationally friendly method to generate them.

On tackling reverse convex constraints for non-overlapping of unequal circles

We study the unequal circle-circle non-overlapping constraints, a form of reverse convex constraints that often arise in optimization models for cutting and packing applications. The feasible region induced by the intersection of circle-circle non-overlapping constraints is highly non-convex, and standard approaches to construct convex relaxations for spatial branch-and-bound global optimization of such models typically yield unsatisfactory loose relaxations. Consequently, solving such non-convex models to guaranteed optimality remains extremely challenging even for the state-of-the-art codes. In this paper, we apply a purpose-built branching scheme on non-overlapping constraints and utilize strengthened intersection cuts and various feasibility-based tightening techniques to further tighten the model relaxation. We embed these techniques into a branch-and-bound code and test them on two variants of circle packing problems. Our computational studies on a suite of 75 benchmark instances yielded, for the first time in the open literature, a total of 54 provably optimal solutions, and it was demonstrated to be competitive over the use of the state-of-the-art general-purpose global optimization solvers.

When Lift-and-Project Cuts Are Different

In this paper, we present a method to determine if a lift-and-project cut for a mixed-integer linear program is irregular, in which case the cut is not equivalent to any intersection cut from the bases of the linear relaxation. This is an important question due to the intense research activity for the past decade on cuts from multiple rows of simplex tableau as well as on lift-and-project cuts from nonsplit disjunctions. Although it has been known for a while that lift-and-project cuts from split disjunctions are always equivalent to intersection cuts and consequently to such multirow cuts, it has been recently shown that there is a necessary and sufficient condition in the case of arbitrary disjunctions: a lift-and-project cut is regular if, and only if, it corresponds to a regular basic solution of the Cut Generating Linear Program (CGLP). This paper has four contributions. First, we state a result that simplifies the verification of regularity for basic CGLP solutions. Second, we provide a mixed-integer formulation that checks whether there is a regular CGLP solution for a given cut that is regular in a broader sense, which also encompasses irregular cuts that are implied by the regular cut closure. Third, we describe a numerical procedure based on such formulation that identifies irregular lift-and-project cuts. Finally, we use this method to evaluate how often lift-and-project cuts from simple t-branch split disjunctions are irregular, and thus not equivalent to multirow cuts, on 74 instances of the Mixed Integer Programming Library (MIPLIB) benchmarks.

Outer-product-free sets for polynomial optimization and oracle-based cuts

This paper introduces cutting planes that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set $$S\cap P$$ , where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S, we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron. These cuts are generated from convex forbidden zones, or S-free sets, derived from the oracle. We also consider the special case of polynomial optimization. Accordingly we develop a theory of outer-product-free sets, where S is the set of real, symmetric matrices of the form $$xx^T$$ . All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify several families of such sets. These families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently. Computational experiments demonstrate the promise of our approach.

A branch-and-cut algorithm for Mixed-Integer Bilinear Programming

In this paper, we consider the Mixed-Integer Bilinear Programming problem, a widely-used reformulation of the classical mixed-integer quadratic programming problem. For this problem we describe a branch-and-cut algorithm for its exact solution, based on a new family of intersection cuts derived from bilinear-specific disjunctions. We also introduce a new branching rule that is specifically designed for bilinear problems. We computationally analyze the behavior of the proposed algorithm on a large set of mixed-integer quadratic instances from the MINLPlib problem library. Our results show that our method, even without intersection cuts, is competitive with a state-of-the-art mixed-integer nonlinear solver. As to intersection cuts, their extensive use at each branching node tends to slow down the solver for most problems in our test bed, but they are extremely effective for some specific instances.

Partial hyperplane activation for generalized intersection cuts

The generalized intersection cut paradigm is a recent framework for generating cutting planes in mixed integer programming with attractive theoretical properties. We investigate this computationally unexplored paradigm and observe that a key hyperplane activation procedure embedded in it is not computationally viable. To overcome this issue, we develop a novel replacement to this procedure called partial hyperplane activation (PHA), introduce a variant of PHA based on a notion of hyperplane tilting, and prove the validity of both algorithms. We propose several implementation strategies and parameter choices for our PHA algorithms and provide supporting theoretical results. We computationally evaluate these ideas in the COIN-OR framework on MIPLIB instances. Our findings shed light on the the strengths of the PHA approach as well as suggest properties related to strong cuts that can be targeted in the future.

The (not so) trivial lifting in two dimensions

When generating cutting-planes for mixed-integer programs from multiple rows of the simplex tableau, the usual approach has been to relax the integrality of the non-basic variables, compute an intersection cut, then strengthen the cut coefficients corresponding to integral non-basic variables using the so-called trivial lifting procedure. Although of polynomial-time complexity in theory, this lifting procedure can be computationally costly in practice. For the case of two-row relaxations, we present a practical algorithm that computes trivial lifting coefficients in constant time, for arbitrary maximal lattice-free sets. Computational experiments confirm that the algorithm works well in practice.

Intersection cuts for single row corner relaxations

We consider the problem of generating inequalities that are valid for one-row relaxations of a simplex tableau, with the integrality constraints preserved for one or more non-basic variables. These relaxations are interesting because they can be used to generate cutting planes for general mixed-integer problems. We first consider the case of a single non-basic integer variable. This relaxation is related to a simple knapsack set with two integer variables and two continuous variables. We study its facial structure by rewriting it as a constrained two-row model, and prove that all its facets arise from a finite number of maximal $$\left( \mathbb {Z}\times \mathbb {Z}_+\right) $$ -free splits and wedges. The resulting cuts generalize both MIR and 2-step MIR inequalities. Then, we describe an algorithm for enumerating all the maximal $$\left( \mathbb {Z}\times \mathbb {Z}_+\right) $$ -free sets corresponding to facet-defining inequalities, and we provide an upper bound on the split rank of those inequalities. Finally, we run computational experiments to compare the strength of wedge cuts against MIR cuts. In our computations, we use the so-called trivial fill-in function to exploit the integrality of more non-basic variables. To that end, we present a practical algorithm for computing the coefficients of this lifting function.

Approximation of Corner Polyhedra with Families of Intersection Cuts

We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets in $\mathbb{R}^n$. In particular, we look at the problem of characterizing families that approximate the corner polyhedron up to a constant factor, which depends only on $n$ and not the data or dimension of the corner polyhedron. The literature already contains several results in this direction. In this paper, we use the maximum number of facets of lattice-free sets in a family as a measure of its complexity and precisely characterize the level of complexity of a family required for constant factor approximations. As one of the main results, we show that, for each natural number $n$, a corner polyhedron with $n$ basic integer variables and an arbitrary number of continuous nonbasic variables is approximated up to a constant factor by intersection cuts from lattice-free sets with at most $i$ facets if $i> 2^{n-1}$ and that no such approximation is possible if $i \le 2^{n-1}$. When the approximation factor is allowed to depend on the denominator of the fractional vertex of the linear relaxation of the corner polyhedron, we show that the threshold is $i > n$ versus $i \leq n$. The tools introduced for proving such results are of independent interest for studying intersection cuts.

On the use of intersection cuts for bilevel optimization

We address a generic mixed-integer bilevel linear program (MIBLP), i.e., a bilevel optimization problem where all objective functions and constraints are linear, and some/all variables are required to take integer values. We first propose necessary modifications needed to turn a standard branch-and-bound MILP solver into an exact and finitely-convergent MIBLP solver, also addressing MIBLP unboundedness and infeasibility. As in other approaches from the literature, our scheme is finitely-convergent in case both the leader and the follower problems are pure integer. In addition, it is capable of dealing with continuous variables both in the leader and in follower problems—provided that the leader variables influencing follower’s decisions are integer and bounded. We then introduce new classes of linear inequalities to be embedded in this branch-and-bound framework, some of which are intersection cuts based on feasible-free convex sets. We present a computational study on various classes of benchmark instances available from the literature, in which we demonstrate that our approach outperforms alternative state-of-the-art MIBLP methods.

Intersection cuts for convex mixed integer programs from translated cones

We develop a general framework for linear intersection cuts for convex integer programs with full-dimensional feasible regions by studying integer points of their translated tangent cones, generalizing the idea of Balas (1971). For proper (i.e, full-dimensional, closed, convex, pointed) translated cones with fractional vertices, we show that under certain mild conditions all intersection cuts are indeed valid for the integer hull, and a large class of valid inequalities for the integer hull are intersection cuts, computable via polyhedral approximations. We also give necessary conditions for a class of valid inequalities to be tangent halfspaces of the integer hull of proper translated cones. We also show that valid inequalities for non-pointed regular translated cones can be derived as intersection cuts for associated proper translated cones under some mild assumptions.

On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts

We examine the connections between the classes of cuts in the title. We show that lift-and-project (L&P) cuts from a given disjunction are equivalent to generalized intersection cuts from the family of polyhedra obtained by taking positive combinations of the complements of the inequalities of each term of the disjunction. While L&P cuts from split disjunctions are known to be equivalent to standard intersection cuts (SICs) from the strip obtained by complementing the terms of the split, we show that L&P cuts from more general disjunctions may not be equivalent to any SIC. In particular, we give easily verifiable necessary and sufficient conditions for a L&P cut from a given disjunction D to be equivalent to a SIC from the polyhedral counterpart of D. Irregular L&P cuts, i.e. those that violate these conditions, have interesting properties. For instance, unlike the regular ones, they may cut off part of the corner polyhedron associated with the LP solution from which they are derived. Furthermore, they are not exceptional: their frequency exceeds that of regular cuts. A numerical example illustrates some of the above properties.

Intersection cuts—standard versus restricted

This note is meant to elucidate the difference between intersection cuts as originally defined, and intersection cuts as defined in the more recent literature. It also states a basic property of intersection cuts under their original definition.

A geometric approach to cut-generating functions

The cutting-plane approach to integer programming was initiated more that 40 years ago: Gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and Balas introduced intersection cuts for the corner polyhedron. This line of research was left dormant for several decades until relatively recently, when a paper of Andersen, Louveaux, Weismantel and Wolsey generated renewed interest in the corner polyhedron and intersection cuts. Recent developments rely on tools drawn from convex analysis, geometry and number theory, and constitute an elegant bridge between these areas and integer programming. We survey these results and highlight recent breakthroughs in this area.

Intersection cuts for nonlinear integer programming: convexification techniques for structured sets

We study the generalization of split, k-branch split, and intersection cuts from mixed integer linear programming to the realm of mixed integer nonlinear programming. Constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split, k-branch split, and intersection cuts for several classes of non-polyhedral sets. In particular, we give simple formulas for split cuts for essentially all convex sets described by a single conic quadratic inequality. We also give simple formulas for k-branch split cuts and some general intersection cuts for a wide variety of convex quadratic sets.

The strength of multi-row models

We develop a method for computing facet-defining valid inequalities for any mixed-integer set $$P_J$$ . While our practical implementation does not return only facet-defining inequalities, it is able to find a separating cut whenever one exists. The separator is not comparable in speed with the specific cutting-plane generators used in branch-and-cut solvers, but it is general-purpose. We can thus use it to compute cuts derived from any reasonably small relaxation $$P_J$$ of a general mixed-integer problem, even when there exists no specific implementation for computing cuts with $$P_J$$ . Exploiting this, we evaluate, from a computational perspective, the usefulness of cuts derived from several types of multi-row relaxations. In particular, we present results with four different strengthenings of the two-row intersection cut model, and multi-row models with up to fifteen rows. We conclude that only fully-strengthened two-row cuts seem to offer a significant advantage over two-row intersection cuts. Our results also indicate that the improvement obtained by going from models with very few rows to models with up to fifteen rows may not be worth the increased computing cost.

Intersection cuts from multiple rows: a disjunctive programming approach

We address the issue of generating cutting planes for mixed integer programs from multiple rows of the simplex tableau with the tools of disjunctive programming. A cut from \(q\) rows of the simplex tableau is an intersection cut from a \(q\)-dimensional parametric cross-polytope, which can also be viewed as a disjunctive cut from a \(2^q\)-term disjunction. We define the disjunctive hull of the \(q\)-row problem, describe its relation to the integer hull, and show how to generate its facets. For the case of binary basic variables, we derive cuts from the stronger disjunctions whose terms are equations. We give cut strengthening procedures using the integrality of the nonbasic variables for both the integer and the binary case. Finally, we discuss some computational experiments.

Intersection Cuts with Infinite Split Rank

We consider mixed-integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cut generated from a rational lattice-free polytope L is finite if and only if the integer points on the boundary of L satisfy a certain “2-hyperplane property.” The Dey–Louveaux characterization is a consequence of this more general result.