Experiments with two-row cuts from degenerate tableaux

TL;DR

This paper investigates the effectiveness of two-row cuts from degenerate tableaux in mixed-integer programming, comparing their performance to GMI cuts through experiments that consider reliability and aggressiveness of cut generation.

Contribution

It provides an empirical analysis of two-row cuts' impact on MIP instances, highlighting the importance of experimental setup and cut generator parameters.

Findings

Two-row cuts can dominate split closures in some cases.

The competitiveness of these cuts relative to GMI cuts depends on experimental conditions.

Reliability and aggressiveness settings significantly influence cut effectiveness.

Abstract

There has been a recent interest in cutting planes generated from two or more rows of the optimal simplex tableau. One can construct examples of integer programs for which a single cutting plane generated from two rows dominates the entire split closure. Motivated by these theoretical results, we study the effect of adding a family of cutting planes generated from two rows on a set of instances from the MIPLIB library. The conclusion of whether these cuts are competitive with GMI cuts is very sensitive to the experimental setup. In particular, we consider the issue of reliability versus aggressiveness of the cut generators, an issue that is usually not addressed in the literature.

Tables

Table 1. Table 1: Average percentage of gap closed at different depths

	Depth 0 (root node)	Depth 4	Depth 8	Depth 12
G	28.33	57.41	68.15	75.23
G+Allpairs	29.11	58.13	68.43	74.81
G+Deepest	28.80	58.09	68.59	75.40
G-2Rounds	36.66	59.41	68.75	75.47

Table 2. Table 2: Average percentage of gap closed at different depths

	Depth 0 (root node)	Depth 4	Depth 8	Depth 12
CglGomory	23.83	51.86	62.91	69.83
CglGomory+All pairs	24.72	53.18	64.37	71.68
CglGomory+Deepest	24.45	53.92	64.54	71.19
CglGomory-2Rounds	30.89	53.73	62.85	69.97

Equations

$$\begin{array}[]{rrcl}&x&=&f+\displaystyle\sum_{j=1}^{k}r^{j}s_{j}\\[2.84544pt] &x&\in&\mathbb{Z}^{2}\\[2.84544pt] &s&\in&\mathbb{R}_{+}^{k}\ .\end{array}$$

$$\sum _{j=1}^k\psi (r^j)s_j\ge 1,$$

$$B_{\psi }:=\{x\in {\mathbb{R}}^2:\psi (x-f)\le 1\}$$

$$x=f+\sum _{j=1}^k{r}^j{s}_j$$

$$\begin{array}[]{rrcl}&x_{i}&=&f_{i}+\displaystyle\sum_{j=1}^{k}r_{i}^{j}s_{j}\\[2.84544pt] &x_{\ell}&=&f_{\ell}+\displaystyle\sum_{j=1}^{k}r_{\ell}^{j}s_{j}\end{array}$$

$$p^2=\left(-1,f_{\ell }-\underset{j}{\min }\left\{\frac{r_{\ell }^j}{r_i^j}|r_i^j<0\right\}\right)\text{mbox}and{p}^3=\left(-1,f_{\ell }-\underset{j}{\max }\left\{\frac{r_{\ell }^j}{r_i^j}|r_i^j<0\right\}\right).$$

$$\left(-x_i\ge 1\right)\vee \left(\eta x_i+x_{\ell }\ge 1\right)\vee \left(\mu x_i-x_{\ell }\ge 0\right)$$

$$\sum _{j=1}^k\max \left\{\frac{-r_i^j}{1+f_i},\frac{\eta r_i^j+r_{\ell }^j}{1-\eta f_i-f_{\ell }},\frac{\mu r_i^j-r_{\ell }^j}{-\mu f_i+f_{\ell }}\right\}{s}_j\ge 1.$$

$$\begin{array}[]{rrcl}&x_{i}&=&f_{i}+\displaystyle\sum_{j=1}^{k}r_{i}^{j}s_{j}\\[2.84544pt] &x_{\ell}&=&f_{\ell}+\displaystyle\sum_{j=1}^{k}r_{\ell}^{j}s_{j}\\[2.84544pt] &x_{i},x_{\ell}&\in&\mathbb{Z},\\[2.84544pt] &s_{j}&\in&\mathbb{Z}\;\;\;\;\mbox{for \ all \ }j\in I\subseteq\{1,\ldots,k\},\\[2.84544pt] &s_{j}&\geq&0\ .\end{array}$$

$${\Phi }_1^j(m_1)$$

$${\Phi }_2^j(m_1,m_2)$$

$${\Phi }_3^j(m_1,m_2)$$

$${\Phi }^j(m_1,m_2)=\max \left\{{\Phi }_1^j(m_1),{\Phi }_2^j(m_1,m_2),{\Phi }_3^j(m_1,m_2)\right\}.$$

$$\sum _{j\in I}\underset{(m_1,m_2)\in {\mathbb{Z}}^2}{\min }{\Phi }^j(m_1,m_2)s_j+\sum _{j\in \bar{I}}{\Phi }^j(0,0)s_j\ge 1,$$

$$\underset{\begin{array}{c}{m}_1,m_2\in \mathbb{Z}\\ m_1\le r_i^j\end{array}}{\min }{\Phi }^j(m_1,m_2)=\min \left\{{\Phi }_2^j\left(\lfloor r_i^j\rfloor ,\lfloor {\overline{m}}_2^j\rfloor \right),{\Phi }_3^j\left(\lfloor r_i^j\rfloor ,\lceil {\overline{m}}_2^j\rceil \right)\right\};$$

$$\underset{\begin{array}{c}{m}_1,m_2\in \mathbb{Z}\\ m_1>r_{1j},m_2\le r_{\ell }^j\end{array}}{\min }{\Phi }^j(m_1,m_2)=\{\begin{array}{ll}\min \left\{{\Phi }_1^j\left(\lceil {\overline{m}}_1^j\rceil \right),{\Phi }_2^j\left(\lfloor {\overline{m}}_1^j\rfloor ,\lfloor r_{\ell }^j\rfloor \right)\right\}& \text{if }{\overline{m}}_1^j>r_i^j,\\ {\Phi }_2^j(\lceil r_i^j\rceil ,\lfloor r_{\ell }^j\rfloor )& \text{otherwise.}\end{array};$$

$$\underset{\begin{array}{c}{m}_1,m_2\in \mathbb{Z}\\ m_1>r_i^j,m_2\ge r_{\ell }^j\end{array}}{\min }\Phi (m_1^j,m_2^j)=\{\begin{array}{ll}\min \left\{{\Phi }_1^j\left(\lceil {\hat{m}}_1^j\rceil \right),{\Phi }_3^j\left(\lfloor {\hat{m}}_1^j\rfloor ,\lceil r_{\ell }^j\rceil \right)\right\}& \text{if }{\hat{m}}_1^j>r_i^j,\\ {\Phi }_3^j\left(\lceil r_i^j\rceil ,\lceil r_{\ell }^j\rceil \right)& \text{otherwise.}\end{array}.$$

$$\frac{\eta \left(r_i^j-m_1\right)+r_{\ell }^j-m_2}{1-\eta f_1-f_2}\ge \frac{\eta \left(r_i^j-\lfloor r_i^j\rfloor \right)+r_{\ell }^j-m_2}{1-f_2-\eta f_1}$$

$$\frac{\mu \left(r_i^j-m_1\right)-r_{\ell }^j+m_2}{f_2-\mu f_1}\ge \frac{\mu \left(r_i^j-\lfloor r_i^j\rfloor \right)-r_{\ell }^j+m_2}{f_2-\mu f_1}$$

$$\min \{{\Phi }^j(m_1,m_2)|m_1,m_2\in \mathbb{Z},m_1\le r_i^j\}=\min \left\{{\Phi }_2^j(\lfloor r_i^j\rfloor ,\lfloor {\overline{m}}_2^j\rfloor ),{\Phi }_3^j(\lfloor r_i^j\rfloor ,\lceil {\overline{m}}_2^j\rceil )\right\}.$$

$$\min \{\Phi (m_1,m_2)|m_1,m_2\in \mathbb{Z},m_1\le r_i^j,m_2\le r_{\ell }^j\}=\min \{\Phi (m_1,\lfloor r_{\ell }^j\rfloor )|m_1\in \mathbb{Z},m_1\le r_i^j\}.$$

$$\begin{array}[]{rrcl}&x_{i}&=&f_{i}+\displaystyle\sum_{j=1}^{k}r_{i}^{j}s_{j}\\[2.84544pt] &x_{\ell}&=&f_{\ell}+\displaystyle\sum_{j=1}^{k}r_{\ell}^{j}s_{j}\\[2.84544pt] &x_{i},x_{\ell}&\in&\mathbb{Z},\\[2.84544pt] &x_{i}&\geq&0\\[2.84544pt] &s_{j}&\in&\mathbb{Z}\;\;\;\;\mbox{for \ all \ }j\in I\subseteq\{1,\ldots,k\},\\[2.84544pt] &s_{j}&\geq&0\ .\end{array}$$

$$\left(\eta x_i+x_{\ell }\ge 1\right)\vee \left(\mu x_i-x_{\ell }\ge 0\right).$$

$$\sum _{j\in J}\max \left\{\frac{\eta r_i^j+r_{\ell }^j}{1-f_{\ell }-\eta f_i},\frac{\mu r_i^j-r_{\ell }^j}{f_{\ell }-\mu f_i}\right\}{s}_j\ge 1.$$

$$\left(\eta \left(x_i+\sum _{j\in I}{m}_j^1{s}_j\right)+x_{\ell }-\sum _{j\in I}{m}_2^j{s}_j\ge 1\right)\vee \left(\mu \left(x_i+\sum _{j\in I}{m}_j^1{s}_j\right)-x_{\ell }+\sum _{j\in I}{m}_2^j{s}_j\ge 0\right)$$

$${\Psi }^j(m_1,m_2)=\max \left\{\frac{\eta \left(r_i^j+m_1\right)+r_{\ell }^j-m_2}{1-f_{\ell }-\eta f_i},\frac{\mu \left(r_i^j+m_1\right)+m_2-r_{\ell }^j}{f_{\ell }-\mu f_i}\right\}.$$

$$\sum _{j\in I}\underset{(m_1,m_2)\in {\mathbb{Z}}_{+}\times \mathbb{Z}}{\min }{\Psi }^j(m_1,m_2)s_j+\sum _{j\in J\setminus I}\max \left\{\frac{\eta r_i^j+r_{\ell }^j}{1-f_{\ell }-\eta f_i},\frac{\mu r_i^j-r_{\ell }^j}{f_{\ell }-\mu f_i}\right\}{s}_j\ge 1$$

$$\underset{(m_1,m_2)\in {\mathbb{Z}}_{+}\times \mathbb{Z}}{\min }{\Psi }^j(m_1,m_2)=\min \left\{\frac{\eta r_i^j+r_{\ell }^j-\lfloor {\overline{m}}_2^j\rfloor }{1-f_{\ell }-\eta f_i},\frac{\mu r_i^j+\lceil {\overline{m}}_2^j\rceil -r_{\ell }^j}{f_{\ell }-\mu f_i}\right\}$$

$$\frac{\eta \left(r_i^j-m_1\right)+r_{\ell }^j-m_2}{1-\eta f_i-f_{\ell }}\ge \frac{\eta r_i^j+r_{\ell }^j-m_2}{1-f_{\ell }-\eta f_i}$$

$$\frac{\mu \left(r_i^j-m_1\right)-r_{\ell }^j+m_2}{f_{\ell }-\mu f_i}\ge \frac{\mu r_i^j-r_{\ell }^j+m_2}{f_{\ell }-\mu f_i}$$

$$\underset{(m_1,m_2)\in {\mathbb{Z}}_{+}\times \mathbb{Z}}{\min }{\Psi }^j(m_1,m_2)=\underset{m_2\in \mathbb{Z}}{\min }{\Psi }^j(0,m_2).$$

$$\Pi =\left({\pi }^{1}x\ge {\pi }_0^1\right)\vee \dots \vee \left({\pi }^{p}x\ge {\pi }_0^p\right)$$

Full text

11footnotetext: Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213.22footnotetext: Supported by a Mellon Fellowship.33footnotetext: LIF, Faculté des Sciences de Luminy, Université de Marseille, France.44footnotetext: Supported by ANR grant ANR06-BLAN-0375.55footnotetext: Supported by NSF grant CMMI0653419, ONR grant N00014-03-1-0133 and ANR grant ANR06-BLAN-0375.66footnotetext: Supported by ONR grant N00014-03-1-0133.

1 Introduction

In the last 15 years, generic cutting planes have played a major role in the progress of mixed integer linear programming (MILP) solvers. Most cutting plane algorithms available in state of the art solvers rely on cuts that can be derived from a single equation (note that the equation used to derive cuts need not be one of the constraints of the problem; it may be obtained by a linear combination of the constraints). Examples are Gomory Mixed Integer cuts [21], MIR cuts [27], lift-and-project cuts [4], or lifted cover inequalities [16]. This general family of cuts is known as the family of split cuts [12]. A natural extension is to derive cuts from more than one equation, an area of research enjoying a revival in recent years.

The study of the Corner Polyhedron started by Gomory and Johnson [22, 23, 25], and of Intersection Cuts introduced by Balas [3] provides a framework for generating cutting planes using multiple equations. Several papers give theoretical results on cuts that can be derived from two equations [1, 6, 8, 15, 17]. In [6], we provide examples of integer programs with two constraints and two integer variables, where the integrality gap can be closed by a single inequality derived from the two constraints, while the value of the linear relaxation obtained by adding all split cuts is arbitrarily close to 0. These examples suggest that certain cuts derived from two equations might improve the efficiency of MILP solvers. In this paper, we test the empirical effectiveness of this class of two-row cuts.

The paper is organized as follows. In Section 2, we recall basic results on two-row cuts and motivate the choice of the triangles used in our separation algorithm. In Section 3, we give closed form formulas for certain types of cuts, discuss integer lifting of these cuts, as well as their strengthening when the basic variables are nonnegative. In Section 4, we present computational results. We show that the conclusions are very sensitive to the experimental setup. We present two different setups which lead to opposite conclusions and argue that the setup based on CglGomory is flawed because it overlooks the issues of reliability versus aggressiveness of the cut generator.

2 Two-Row Cutting Planes

2.1 Basic Theory

In this paper, we study a class of cutting planes derived from two equations. Consider a mixed-integer problem with two free integer variables and a finite number of nonnegative, continuous variables.

[TABLE]

We assume $f\in\mathbb{Q}^{2}\setminus\mathbb{Z}^{2}$, $k\geq 1$, and for $j=1,\ldots,k$, $r^{j}\in\mathbb{Q}^{2}\setminus\left\{0\right\}$.

Model (1) naturally arises as a relaxation of a MILP having a basic feasible solution of its linear relaxation with at least two basic integer variables, at least one of which takes a fractional value in the optimal solution. Indeed, consider the simplex tableau corresponding to the basic solution and keep only two equations corresponding to these two basic integer variables. Model (1) was introduced by Andersen, Louveaux, Weismantel, and Wolsey [1]. Several recent papers discuss some of its theoretical properties such as characterization of facets and relative strength of classes of facets  [8, 6, 15, 17] but little empirical evidence on the strength of the resulting cuts is available [19]. The remainder of this section is a short summary of results obtained in [8, 6, 15, 17] that are directly relevant to the work in this paper.

The inequalities $s_{j}\geq 0$ are called trivial for (1). A nontrivial valid inequality for (1) is of the form

[TABLE]

where $\psi:\mathbb{R}^{2}\rightarrow\mathbb{R}_{+}$. A nontrivial valid inequality is minimal if there is no other nontrivial valid inequality $\sum_{j=1}^{k}\psi^{\prime}(r^{j})s_{j}\geq 1$ such that $\psi^{\prime}(r^{j})\leq\psi(r^{j})$ for all $j=1,\ldots,k$. Minimal nontrivial valid inequalities are associated with functions $\psi$ that are nonnegative positively-homogeneous piecewise-linear and convex. Furthermore, the set

[TABLE]

is a maximal lattice-free convex set with $f$ in its interior [8]. By lattice-free convex set we mean a convex set with no integral point in its interior. However integral points are allowed on the boundary. These maximal lattice-free convex sets are splits, triangles, and quadrilaterals as proved by Lovász [26]. In this paper, we will focus on splits and triangles only.

A split is a set of the form $c\leq ax_{1}+bx_{2}\leq c+1$ where $a$ and $b$ are coprime integers and $c$ is an integer.

Following Dey and Wolsey [17], we partition the maximal lattice-free triangles into three types (see Figure 1):

• •

Type 1 triangles: triangles with integral vertices and exactly one integral point in the relative interior of each edge;

• •

Type 2 triangles: triangles with at least one fractional vertex $v$, exactly one integral point in the relative interior of the two edges incident to $v$ and at least two integral points on the third edge;

• •

Type 3 triangles: triangles with exactly three integral points on the boundary, one in the relative interior of each edge.

Figure 1 shows these three types of triangles as well as a maximal lattice-free quadrilateral and a split.

In this paper we focus on simple classes of split and Type 2 triangles. We motivate our choice of cuts in Section 2.2. In Section 3, we give closed form formulas for our cuts. These cuts can be seen as a particular case of intersection cuts [3].

2.2 Type 2 triangles versus splits

For model (1), the split closure is defined as the intersection of all minimal valid inequalities obtained from splits [12]. Similarly, the Type 2 triangle closure is the intersection of all the minimal valid inequalities generated from Type 2 triangles. The Type 2 triangle closure approximates the convex hull of (1) within a factor of 2 (see [6] Theorem 7.3 for a precise statement). On the other hand, the split closure is not always a good approximation of the convex hull of (1). In [6], we construct integer programs where a single inequality derived from a Type 2 triangle closes the integrality gap entirely whereas the split closure closes an arbitrarily small fraction of the gap. In these examples, $f$ has one integral and one fractional component.

These results suggest that it might be interesting to generate cuts from Type 2 triangles in addition to the classical Gomory Mixed Integer cuts [30] when solving an MILP. Unfortunately, it appears difficult to separate over all Type 2 triangle cuts, as it seems to be at least as hard as the NP-hard problem of separating over the split closure [9, 14]. In this paper, we separate a very restricted class of Type 2 triangle cuts. Consider the simplex tableau obtained by solving the LP relaxation of MILP. We have

[TABLE]

where $x$ denotes the basic variables and $s$ the nonbasic. Motivated by the examples mentioned above, we test the effectiveness of cutting planes from two rows of the simplex tableau

[TABLE]

where $x_{i}$ and $x_{\ell}$ are basic integer variables, $f_{i}$ is integral and $f_{\ell}$ is fractional, i.e. at least one of the basic integral variables has an integral value and at least one has a fractional value. Note that this will typically occur when the basis is degenerate. Without loss of generality we can make a translation so that $f_{i}=0$ and $f_{\ell}\in]0,1[$. We try to construct two Type 2 triangles, one where the edge that contains at least two integral points is on the line $x_{i}=-1$, and a symmetrical one where this edge is on the line $x_{i}=1$. As these two cases are similar, we focus on the first one in the following discussion. For each ray $r^{j}$ such that the half-line $f+\lambda r^{j}$ for $\lambda>0$ intersects the plane $x_{i}=-1$, we compute that intersection and its projection onto the $(x_{i},x_{\ell})$-plane. Among these projections, we define $p^{2}=(p^{2}_{i},p^{2}_{\ell})$ (resp. $p^{3}=(p^{3}_{i},p^{3}_{\ell})$) as the point with largest (resp. smallest) $x_{\ell}$-coordinate, i.e.,

[TABLE]

Note that $p^{2}$ and $p^{3}$ are not defined if $r^{i}_{j}\geq 0$ for all $j=1,\ldots,k$. In this case or when $p^{2}=p^{3}$, we do not generate a cut. Otherwise, we consider three cases:

• i)

There are at least two integral points in the interior of the segment $p^{2}p^{3}$. We construct a Type 2 triangle with vertices $p^{1}$, $p^{2}$ and $p^{3}$, where $p^{1}$ is the intersection of the lines passing through $p^{2}$ and $(0,1)$ and through $p^{3}$ and $(0,0)$.

• ii)

There is exactly one integral point in the interior of the segment $p^{2}p^{3}$. Let $q^{2}=(-1,\lceil p^{2}_{\ell}\rceil)$, and $q^{3}=(-1,\lfloor p^{3}_{\ell}\rfloor)$. If $\lceil p^{2}_{\ell}\rceil-p^{2}_{\ell}\leq p^{3}_{\ell}-\lfloor p^{3}_{\ell}\rfloor$, we construct a Type 1 or Type 2 triangle whose vertices are $p^{1}$, $q^{2}$ and $p^{3}$ where $p^{1}$ is the intersection of the lines passing through $q^{2}$ and $(0,1)$ and through $p^{3}$ and $(0,0)$. Otherwise, we construct a Type 2 triangle whose vertices are $p^{1},p^{2}$ and $q^{3}$, where $p^{1}$ is the intersection of the lines passing through $p^{2}$ and $(0,1)$ and through $q^{3}$ and $(0,0)$.

• iii)

There is no integral point in the interior of the segment $p^{2}p^{3}$. Let $q^{2}=(-1,\lceil p^{2}_{\ell}\rceil)$, and $q^{3}=(-1,\lfloor p^{3}_{\ell}\rfloor)$. We construct a split whose sides are the lines through $q^{2}$ and $(0,1)$ and through $q^{3}$ and $(0,0)$.

Therefore, if the optimal tableau of the LP relaxation of the MILP has $m$ basic integer variables with an integer value and $n$ basic integer variables that are fractional, this algorithm generates up to $2nm$ cuts from the optimal simplex tableau. We use variants of this basic algorithm for generating and testing two-row cuts.

3 Derivation of the cuts

3.1 Formulas of the cut for Type 1 and Type 2 triangles

Consider a lattice-free triangle $T$ of Type 1 or 2 with vertices $p^{1}$, $p^{2}$ and $p^{3}$. Without loss of generality, assume that $T$ is in a canonical form where $p^{2}=(-1,1+\eta)$ and $p^{3}=(-1,-\mu)$ and $0\leq\eta\leq\mu\leq\eta+1$. Furthermore we assume that the points $(0,0)$ and $(0,1)$ are respectively in the relative interior of edge $p^{3}p^{1}$ and $p^{2}p^{1}$. A Type 1 or Type 2 triangle can always be put into this canonical form by applying a unimodular transformation (see Lemma 11 and Proposition 26 in [17]).

For a Type 1 triangle, we thus have $\eta=0$ and $\mu=1$ and the three vertices of the triangle are $p^{1}=(1,0)$, $p^{2}=(-1,2)$ and $p^{3}=(-1,0)$, see Figure 2. For a Type 2 triangle the points $(-1,0)$ and $(-1,1)$ are in the relative interior of the line segment $p^{2}p^{3}$, see Figure 2.

As noted in [1], the cut obtained from the lattice-free set $T$ is the intersection cut [3] derived from the disjunction

[TABLE]

Disjunction (5) simply states that all feasible points are not in the interior of triangle $T$. Using the formula of the intersection cut (see Appendix A), the intersection cut obtained from the disjunction (5) is

[TABLE]

Since $T$ is a maximal lattice-free triangle, (6) is a minimal inequality for (1). In the next two sections, we show how the cut (6) can be strengthened if some of the non-basic variables are integer constrained (this operation is called an integer lifting of the cut) or if $x_{i}$ is nonnegative.

3.2 Integer Lifting

In this section, we consider a variation of the two-row problem (4) where some of the variables $s_{j}$ are constrained to be integer

[TABLE]

Clearly inequality (6) is valid for (7). Dey and Wolsey [17] show how to strengthen the coefficients of the non-basic integer variables in (6) to obtain a minimal inequality for (7). This strengthening is called integer lifting. If the lattice-free set used to generate the cut is a triangle of Type 1 or 2, Dey and Wolsey [17] show that there is a unique integer lifting that produces a minimal inequality for (7) and it is obtained through the so-called trivial fill-in function. In this section, we derive a closed-form formula for this trivial lifting. We use the terminology of intersection cuts [3].

To derive the formula of the cut strengthened by using the trivial fill-in function (see Appendix B), we define for any $j\in\{1,\ldots,k\}$ and any integers $m_{1}$ and $m_{2}$:

[TABLE]

and

[TABLE]

The strengthened cut is then given by:

[TABLE]

where $\bar{I}=\{1,\ldots,k\}\setminus I$. The following proposition gives the closed form formula for $\min_{(m_{1},m_{2})\in\mathbb{Z}^{2}}\Phi^{j}(m_{1},m_{2})$.

Proposition 3.1**.**

Let $\overline{m}^{j}_{2}=r^{j}_{\ell}+\frac{\left(r^{j}_{i}-\lfloor r^{j}_{i}\rfloor\right)\left(-\mu+\left(\mu+\eta\right)f_{\ell}\right)}{1-\left(\mu+\eta\right)f_{i}}$, $\overline{m}^{j}_{1}=r^{j}_{i}-\frac{(1+f_{i})\left(r^{j}_{\ell}-\lfloor r^{j}_{\ell}\rfloor\right)}{1+\eta-f_{\ell}}$ and $\widehat{m}^{j}_{1}=r^{j}_{i}-\frac{(1+f_{i})(\lceil r^{j}_{\ell}\rceil-r^{j}_{\ell})}{\mu+f_{\ell}}$.

1. (i)

[TABLE] 2. (ii)

[TABLE] 3. (iii)

[TABLE]

Proof.

(i) First we show that if $m_{1}\leq r^{j}_{i}$, then $\Phi^{j}(m_{1},m_{2})=\max\left\{\Phi^{j}_{2}(m_{1},m_{2}),\Phi^{j}_{3}(m_{1},m_{2})\right\}.$ Note that if $m_{1}\leq r^{j}_{i}$, $\Phi^{j}_{1}(m_{1})=\frac{m_{1}-r^{j}_{i}}{1+f_{1}}\leq 0$. On the other hand, since $\eta$ and $\mu$ are nonnegative and either $m_{2}-r^{j}_{\ell}$ or $r^{j}_{\ell}-m_{2}$ is nonnegative $\Phi^{j}(m_{1},m_{2})\geq 0$.

Since $\eta$ and $\mu$ are positive, if $m_{1}\leq\lfloor r^{j}_{i}\rfloor$:

[TABLE]

and therefore $\min\{\Phi^{j}(m_{1},m_{2})\ |\ m_{1},m_{2}\in\mathbb{Z},m_{1}\leq r^{j}_{i}\}=\min\{\Phi^{j}(\lfloor r^{j}_{i}\rfloor,m_{2})\ |\ m_{2}\in\mathbb{Z}\}$. This last minimum is attained either in $\Phi^{j}(\lfloor r^{j}_{i}\rfloor,\lceil\overline{m}_{2}\rceil)$ or $\Phi^{j}(\lfloor r^{j}_{i}\rfloor,\lfloor\overline{m}_{2}\rfloor)$. Therefore

[TABLE]

(ii) We now suppose that $m_{1}>r^{j}_{i}$ and $m_{2}\leq r^{j}_{\ell}$. Then $\Phi^{j}_{3}(m_{1},m_{2})\leq 0$ and therefore $\Phi^{j}(m_{1},m_{2})=\max\left\{\Phi^{j}_{1}\left(m_{1}\right),\Phi^{j}_{2}\left(m_{1},m_{2}\right)\right\}$. Furthermore, we have:

[TABLE]

If $\overline{m}_{1}<r^{j}_{i}$, the minimum is attained either in $\Phi^{j}(\lfloor\overline{m}_{1}\rfloor,\lfloor r^{j}_{l}\rfloor)$ or $\Phi(\lceil\overline{m}_{1}\rceil,\lfloor r^{j}_{l}\rfloor)$, otherwise the minimum is attained in $\Phi^{j}(\lceil r^{j}_{i}\rceil,\lfloor r^{j}_{l}\rfloor)$. The formula follows.

(iii) Similar to the proof of (ii). ∎

3.3 Using nonnegativity of basic variables

In model (1), the two basic variables are assumed to be free, but it may happen that one or both are constrained to be nonnegative in the original model. This model was studied by Fukasawa and Günlük [20], Dey and Wolsey [17], and Conforti, Cornuéjols and Zambelli [10].

In this section, we consider the case where $x_{i}$ is constrained to be nonnegative, and provide a closed-form formula to strengthen the cuts (6).

Intuitively, for a triangle as described in Section 2, if we have $x_{i}\geq 0$ then we can remove the vertical side of the triangle and expand it into an unbounded set containing no feasible solution of (1). We call this set a wedge.

We consider a Type 1 or 2 triangle in canonical form together with the feasible set, including the nonnegativity on $x_{i}$:

[TABLE]

The following disjunction is valid

[TABLE]

The intersection cut obtained from (9) is given by:

[TABLE]

Note that the intersection cut (10) dominates (6). In particular some of the coefficients of the latter cut may be negative. Balas [2] observed the following.

Proposition 3.2**.**

Inequality (10) is valid for the split closure.

Proof.

Let $P$ be the polyhedron obtained by dropping the integrality constraints in (8). Let $P_{1}=P\cap\{x_{\ell}\geq 1\}$ and $P_{2}=P\cap\{x_{\ell}\leq 0\}$. Inequality (10) is valid for $\left(P\cap\{\eta x_{i}+x_{\ell}\geq 1\}\right)\cup\left(P\cap\{\mu x_{i}-x_{\ell}\geq 0\}\right)$. Since $P_{1}\subseteq P\cap\{\eta x_{i}+x_{\ell}\geq 1\}$ and $P_{1}\subseteq P\cap\{\mu x_{i}-x_{\ell}\geq 0\}$, inequality (10) is valid for $P_{1}\cup P_{2}$, which proves that it is a split inequality. ∎

An optimal integer lifting for this cut can be computed in a similar way to the trivial lifting computed in Section 3.2. The following proposition describes how the lifting can be adapted for disjunction (9). The difference with the lifting performed using Proposition B.1 is that here the disjunction can be shifted by any integer in the direction of $x_{\ell}$ but only by a positive integer in the direction of $x_{i}$. See [10] for a proof that this lifting is the unique minimal lifting.

Proposition 3.3**.**

For $j\in I$, let $m^{j}\in\mathbb{Z}^{2}$ with $m^{j}_{1}\geq 0$. The disjunction

[TABLE]

is satisfied by all points $((x_{i},x_{\ell}),s)\in\mathbb{Z}^{2}\times\mathbb{R}^{n}$ with $x_{i}\geq 0$ and $s_{j}\in\mathbb{Z}_{+}$ $\forall j\in I$.

Proof.

Suppose that $((x_{i},x_{\ell}),s)\in\mathbb{Z}^{2}\times\mathbb{R}^{n}$ with $x_{i}\geq 0$ and $s_{j}\in\mathbb{Z}$ $\forall j\in I$ does not satisfy (11). Let $x^{\prime}_{i}=x_{i}+\sum_{j\in I}m^{j}_{1}s_{j}$ and $x^{\prime}_{\ell}=x_{\ell}-\sum_{j\in I}m^{j}_{2}s_{j}$. Then $x^{\prime}_{i}\in\mathbb{Z}_{+}$ and $x^{\prime}_{\ell}\in\mathbb{Z}$ but $(x^{\prime}_{i},x^{\prime}_{\ell})$ does not satisfy (9). ∎

Using this proposition, we can derive a formula for a strengthened cut which uses the integrality of the variables $s_{j}$ for $j\in I$. For $m_{1}\in\mathbb{Z}_{+}$ and $m_{2}\in\mathbb{Z}$, we define

[TABLE]

The following cut is valid by application of Proposition 3.3:

[TABLE]

The next proposition gives a closed form formula for computing the coefficients of this strengthened cut.

Proposition 3.4**.**

[TABLE]

with $\overline{m}^{j}_{2}=r^{j}_{\ell}+\frac{r^{j}_{i}\left(-\mu+\left(\mu+\eta\right)f_{\ell}\right)}{1-\left(\mu+\eta\right)f_{i}}$

Proof.

Since $\eta$ and $\mu$ are positive and $m_{1}\geq 0$

[TABLE]

and therefore

[TABLE]

The minimum is attained either in $\Psi^{j}(0,\lfloor\overline{m}^{j}_{2}\rfloor)$ or $\Psi^{j}(0,\lceil\overline{m}^{j}_{2}\rceil)$. ∎

Note that the cuts generated from the liftings of Propositions 3.1 and 3.4 are incomparable, i.e. neither is guaranteed to dominate the other. They will be compared in Section 4.

4 Computational tests

We now present computational results obtained by applying the separation procedure devised in Section 2.2. The cut generator is implemented in C++ using the COIN-OR framework [33] with Cbc-2.2.2 and using Clp-1.8.2 for solving linear programs. The machine used for all the experiments in this section is a 64 bit Monarch Empro 4-Way Tower Server with four AMD Opteron 852 2.6GHz processors, each with eight DDR-400 SDRAM of 2 GB and running Linux Fedora 11. The compiler is gcc version 4.4.0 20090506 (Red Hat 4.4.0-4). Results are obtained using only one processor. Test instances for our experiments are the sixty-eight MIPLIB3_C_V2 instances [34]. These instances are slight modifications of the standard MIPLIB3 [7] instances for which the validity of a provided feasible solution can be checked in finite precision arithmetic.

The main motivation behind these tests is to determine if triangle cuts are substantially different from traditional Gomory Mixed Integer (GMI) cuts, and if using them in conjunction with GMI cuts might be useful. In other words, we are not suggesting to replace GMI cuts by triangle cuts. Rather, we want to investigate if triangle cuts can improve performance when used in addition to GMI cuts.

4.1 Results on gap closed by our cuts

We compare the following four generators. As described in Section 2.2, we partition the rows of the optimal simplex tableau corresponding to the basic integer variables into fractional rows and integer rows.

• •

G : A cut generator which generates one GMI cut for each fractional row.

• •

G-2rounds : A cut generator which applies G to generate a first round of GMI cuts, reoptimizes the resulting LP and generates a second round of GMI cuts from the new optimal basic solution.

• •

G+Allpairs : A cut generator which generates one GMI cut for each fractional row and all cuts derived from every pair of fractional and integer rows as explained in Section 2.2. We also add cuts derived by removing the vertical side of the triangle whenever possible, as outlined in Section 3.3.

• •

G+Deepest : G+Allpairs often generates a large number of cuts. To reduce this number, we only add the GMI cuts and the deepest among all possible triangle cuts for each fractional row. Note that G generates one cut per fractional row, whereas G+Deepest generates two.

Since computers work in finite precision, cut generators sometimes generate invalid cuts. To limit these occurrences, a variety of tolerances and safeguards are typically used. For GMI cuts, the most important ones are a lower bound on integer infeasibility and an upper bound on dynamism. For a solution $\bar{x}$, define the integer infeasibility of an integer variable $x_{i}$ as $\min\{\bar{x}_{i}-\lfloor\bar{x}_{i}\rfloor,\lceil\bar{x}_{i}\rceil-\bar{x}_{i}\}$. We do not generate a cut from a fractional row whose basic integer variable has an integer infeasibility smaller than 0.01. Define the dynamism of a cut as the ratio between the largest and the smallest nonzero absolute values of its left hand side coefficients. We discard generated cuts whose dynamism is larger than $10^{9}$. We use the same parameters for the generation of two-row cuts with, in addition, the condition that, for a row to be considered an integer row, the integer infeasibility of its basic integer variable must be at most $10^{-5}$.

A method for testing the accuracy and strength of cut generators is given in [28]. Next we give a short description of the method. Please refer to the original paper for additional details.

Consider an instance $I$ with a known feasible solution $x^{*}$. Define a cutting step as the operation of adding cuts obtained from a cut generator and checking if $x^{*}$ is still feasible (report failure of the cut generator if $x^{*}$ is no longer feasible). Define a branching step as getting an optimal solution $\bar{x}$ of the current LP relaxation, picking at random an integer variable $x_{i}$ with a fractional value $\bar{x}_{i}$ and imposing in the LP that its value must be $x^{*}_{i}$.

The method, called a dive towards a feasible solution $x^{*}$, amounts to repeatedly performing a cutting step followed by a branching step until an integer feasible solution of the LP relaxation is obtained. In our experiments $x^{*}$ is chosen to be a known feasible solution whose objective value is likely to be optimum or close to optimum. A number of dives (we use twenty dives in our experiments) are performed for each instance $I$. In our experiments, we set a time limit of 3 hours for each dive.

The motivation for using cuts in a branch-and-cut algorithm is to improve the bound obtained from the linear relaxation and to reduce the size of the enumeration tree. With these two goals in mind, we track the following during each dive:

• •

Fraction of the integrality gap closed after some given numbers of branching steps;

• •

Number of branching steps required to close some given fractions of the integrality gap.

The random branching step allows for meaningful statistical analysis of these two performance measures. We used the nonparametric Quade test as described in [11]. All tests are done with a confidence level of 95%. The statistical package R [36] version 2.10.0 (2009-10-26) is used for the statistical analysis of the results.

In Table 1, we give the average gap closed by the four algorithms G, G-2Rounds, G+Allpairs, G+Deepest after 0, 4, 8 and 12 branching steps in a dive. The following problems timed out without any results for at least one generator: 10teams_c, air04_c, air05_c, arki_c, dano3mip_c, fast0507_c, mitre_c, mkc_c, seymour_c. The table gives results for the remaining 59 instances.

We make the following observations from our results.

• •

Comparing G and G+Allpairs, we note that triangle cuts only improve marginally over the GMI cuts at depths 0, 4 and 8. The improvement in average gap closed at these three depths is less than 1%. At depth 12, the average gap closed is actually more for G, but this difference is not statistically significant according to the Quade test.

Even restricting to instances that are degenerate at the root, the improvement is only 1.2% on average at the root which is disappointingly small given that we are generating at least four times as many cuts.

• •

Comparing G+Allpairs and G+Deepest, we note that most of the above improvement at depths 0, 4 and 8 can be achieved using only the deepest triangles. In fact, surprisingly, at depths 8 and 12 G+Deepest closes more average gap than G+Allpairs, and at depth 12 the difference is statistically significant. This may be due to the fact that adding too many cuts deteriorates the numerical properties of the basis, and this has an adverse effect on the quality of the cuts at greater depths.

• •

Comparing G-2Rounds and G+Deepest, which generate roughly the same number of cuts, we conclude that it is clearly better to use GMI cuts at levels 0 and 4. This is confirmed by the Quade test which shows that the difference is statistically significant. At depths 8 and 12 the difference is not statistically significant.

The results from Table 1 seem to indicate that our family of two-row cuts is not competitive with GMI cuts in terms of gap closed. This conclusion is confirmed by statistical tests performed on the second performance measure introduced above. For brevity, the detailed results of these tests are not reported here.

We performed other experiments which seem to suggest that among the two-row cuts that we generate, the split cuts are the more useful ones. First, we considered two variants of G+Deepest where the choice of the cut is based on the length of the segment $p^{2}p^{3}$ as defined in Section 2.2. In the first variant, we chose the cut with the longest segment $p^{2}p^{3}$; whereas, in the second we chose the shortest. The motivation for the first choice comes from the theoretical results in [6], where it is shown that the gap closed by the split closure is smaller when the segment $p^{2}p^{3}$ is long. However, surprisingly, the second variant closes more gap on average at levels 0, 4, 8 and 12, although the differences are small. This may be explained from the fact that the second variant generates more splits than the first and, in fact, at the root it generates almost only splits.

In another experiment, we tried to see whether our two-row cuts could cut off the optimum vertex of the split closure, as computed by Balas and Saxena [5]. For this purpose we used the formulations in OSCLIB 1.0 [35]. In all but 2 of these instances, we could not cut off the split optimum.

We also compared the depths of the GMI cuts with our two-row cuts. For 15 instances, no triangle cuts are generated at the root because the optimal tableau is not degenerate and there are no basic integer variables with integer values. For the instance nw04_c, the average depths are too small for any meaningful comparison. For each of the 52 remaining instances, we measure the depth of the deepest triangle cut as a percentage of the depth of the GMI cut from the same fractional row. We then average this percentage over all fractional rows to get a single percentage number associated with that instance. For 46 out of the 52 instances, this percentage is between 99 and 125. The other 6 have a percentage between 125 and 200. This shows that, in general, the deepest triangle cuts are deeper than the corresponding GMI cuts.

4.2 Importance of the Experimental Setup

In this section, we show that one can reach drastically different conclusions by modifying the experimental setup in a seemingly natural way. Instead of using our own GMI cut generator, we now use the GMI cut generator CglGomory from the COIN-OR library with default settings. The table corresponding to Table 1 is as follows :

The conclusions we reach from this table are different from those reached from Table 1. We conclude that CglGomory+All pairs is significantly better than CglGomory at all depths and this advantage increases with depth, with the difference being almost 2% at depth 12. These differences are statistically significant according to the Quade test. Moreover, CglGomory+Deepest is superior to CglGomory-2Rounds at depths 4, 8 and 12 with respect to gap closed and this difference is statistically significant at depths 8 and 12. When comparison is made on the number of branching steps needed to close different fractions of the gap, CglGomory+Deepest is significantly better than CglGomory-2Rounds for closing 70% or more of the gap according to the Quade test.

How do we reconcile the different conclusions reached from Tables 1 and 2 ?

Although our goal in this paper is to compare the strength of different cut families, in practice we can only compare cut generators. The generator G described in the previous section and CglGomory are two different generators for GMI cuts. We described above the parameter settings for accepting the cuts in G. For the default CglGomory generator the bound on integer infeasibility is $0.05$, the bound on dynamism is $10^{9}$, and there are several other parameters that may affect the acceptance or rejection of a GMI cut. Even if we set the integer infeasibility and dynamism bounds to the same value in both generators, there are still significant differences in these two cut generators. This raises the question of which generator to use for our experiments and how to set the parameters. Depending on the parameter settings, a cut generator may reject cuts more often (conservative strategy) or less frequently (aggressive strategy).

A typical way that researchers test new cut generators is to add their generator to existing base generators and evaluate the difference in performance. In such a setting, if the base generator is conservative and the tested generator is aggressive, one may expect good results. On the other hand, if the base generator is made more aggressive or the tested generator is made more conservative, these improvements might be eroded to a large extent. The difference between Tables 1 and 2 is a concrete example of this phenomenon. This issue is usually not addressed in the literature. There is a trade-off between reliability of cut generators and their aggressiveness. Therefore, a key aspect of such experimental setups should involve measuring the reliability of cut generators. This was addressed by Margot in  [28]. The general philosophy is to put the generator under stress and to record the number of failures due to invalid cut generation. Concretely, the reliability of a cut generator is tested using the diving experiment described in the previous section where 10 rounds of cuts are generated before a branching step is performed. According to this test, G is comparable in reliability to CglGomory. In fact, G never generated an invalid cut for the 59 instances of MIPLIB3_C_V2 considered in this paper; whereas, CglGomory generated invalid cuts on the instance harp2_c.

In our view, G should be used over CglGomory in our experiments because it is more aggressive and equally reliable. Another advantage of G over CglGomory is that not only are the parameter settings the same as those for the two-row cut generators, the computer code itself is the same. This does not necessarily imply that G should be preferred to CglGomory in a general branch-and-cut setting. More experiments would be needed to assess the reliability of the two cut generators.

This discussion shows that the setup based on CglGomory is flawed and that the correct setup is the one used in Section 4.1, if the goal is to evaluate our two-row cuts as compared to GMI cuts. The conclusion is that our family of two-row cuts is not competitive with GMI cuts.

Appendix A Intersection cuts from multiple rows

Proposition A.1 gives the general formula for the cut derived from a $p$-term disjunction and $m$ rows of a simplex tableau.

Proposition A.1**.**

Let $C=\{(x,s)\in\mathbb{Z}^{m}\times\mathbb{R}^{k}:x=f+\sum_{j=1}^{k}r^{j}s_{j},s\geq 0\}$ and

[TABLE]

be a disjunction satisfied by all points of $\mathbb{Z}^{m}$ and not satisfied by $f$. The inequality

[TABLE]

is valid for $C$.

Proof.

For each $l=1,\ldots,p$, $\pi^{l}x\geq\pi^{l}_{0}$ can be rewritten using the definition of $C$ in terms of the $s$ variables only:

[TABLE]

Reorganizing the terms of the above inequality it can be rewritten as

[TABLE]

The point $f$ does not satisfy the disjunction and therefore $\pi^{l}_{0}-\pi^{l}f>0$, dividing the inequality by $\pi^{l}_{0}-\pi^{l}f$ we obtain

[TABLE]

Each of the above inequalities is valid for one term of the disjunction. Since they all have the same right-hand-side and $s\geq 0$, the inequality obtained by taking the component-wise maximum is valid for all terms of the disjunction and therefore

[TABLE]

is valid for $C$. ∎

The inequality is nothing more than the intersection cut obtained from the cone $C$ and the disjunction $\Pi$.

Appendix B Integer lifting by the trivial fill-in function

Proposition B.1**.**

Let $C=\{(x,s)\in\mathbb{Z}^{m}\times\mathbb{R}^{k}:x=f+\sum_{j=1}^{k}r^{j}s_{j},s\geq 0\}$. Let $I$ be a subset of $\{1,\ldots,k\}$, $C^{I}=C\cap\left\{(x,s)\in\mathbb{Z}^{m}\times\mathbb{R}^{k}:s_{j}\in\mathbb{Z},\,\forall j\in I\right\}$ be a mixed-integer set and

[TABLE]

be a disjunction satisfied by all $x\in\mathbb{Z}^{m}$. For $j\in I$, let $m^{j}$ be an integral vector of dimension $m$. The disjunction

[TABLE]

is valid for $C^{I}$.

Proof.

Suppose that the disjunction $\Pi^{\prime}$ is not valid. Then there exist $(x,s)\in\mathbb{Z}^{m}\times\mathbb{R}^{k}$ with $s_{j}\in\mathbb{Z},\,\forall j\in I$, such that:

[TABLE]

Let $x^{\prime}=x-\sum_{j\in I}m^{j}s_{j}$. $x^{\prime}\in\mathbb{Z}^{m}$ since $x\in\mathbb{Z}^{m}$, $m^{j}\in\mathbb{Z}^{m}$ and $s_{j}\in\mathbb{Z}$ for all $j\in I$. Then $x^{\prime}$ violates disjunction $\Pi$ which contradicts the fact that $\Pi$ is a valid disjunction. ∎

Now we consider the intersection cut obtained from a disjunction of the form $\Pi^{\prime}$ stated in Proposition B.1.

Proposition B.2**.**

Let $C=\{(x,s)\in\mathbb{Z}^{m}\times\mathbb{R}^{k}:x=f+\sum_{j=1}^{k}r^{j}s_{j},s\geq 0\}$. Let $J=\{1,\ldots,k\}$ and $I$ be a subset of $J$, $C^{I}=C\cap\left\{(x,s)\in\mathbb{Z}^{m}\times\mathbb{R}^{k}:s_{j}\in\mathbb{Z},\,\forall j\in I\right\}$ be a mixed-integer set and

[TABLE]

be a disjunction satisfied by all $x\in\mathbb{Z}^{m}$ and not satisfied by the point $f$. For all $m^{j}\in\mathbb{Z}^{m}$, $j=1,\ldots,m$ the inequality

[TABLE]

is valid for $C^{I}$.

Proof.

Since $\Pi$ is valid, for all $m^{j}\in\mathbb{Z}^{m}$, $j=1,\ldots,m$ the disjunction

[TABLE]

is valid.

The terms of the disjunction can be rewritten as:

[TABLE]

The point $f$ does not satisfy the disjunction and therefore $\pi^{l}_{0}-\pi^{l}f>0$, dividing the inequality by $\pi^{l}_{0}-\pi^{l}f$ we obtain

[TABLE]

Each of the above inequalities is valid for one term of the disjunction. By applying, the disjunctive principle we obtain (14). ∎