Intersection cuts for single row corner relaxations

Abstract

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.

Appendices

A Trivial lifting function for S-free sets

In order to simplify the exposition, we examine here the lifting problem in a more general context and adopt the standard approach of the infinite relaxation, as well as its usual notation. We refer the reader to [16] for an introduction.

Let $S := \mathbb {Z}^m \cap Q$, where Q is some rational polyhedron. Let $f \in \mathbb {R}^m \setminus S$. We define

$$\begin{aligned} R_f := \left\{ y \in \mathbb {R}^{\mathbb {R}^m}_+ \; : \; f + \sum _{r \in \mathbb {R}^m} r y_r \in S, \;\; y \text { has a finite support} \right\} , \end{aligned}$$

and a lifted version of $R_f$,

$$\begin{aligned} M_f := \left\{ y \in \mathbb {R}^{\mathbb {R}^m}_+, z \in \mathbb {Z}^{\mathbb {R}^m}_+ \; : \; f + \sum _{r \in \mathbb {R}^m} r y_r + \sum _{r \in \mathbb {R}^m} r z_r \in S, \;\; y, z \text { have a finite support} \right\} . \end{aligned}$$

We say that a function $\psi : \mathbb {R}^m \rightarrow \mathbb {R}$ is valid for $R_f$ if $\sum _{r \in \mathbb {R}^m} \psi (r) y_r \ge 1$ for all $y \in R_f$. We say that $\psi , \pi : \mathbb {R}^m \rightarrow \mathbb {R}$ is valid for $M_f$ if $\sum _{r \in \mathbb {R}^m} \psi (r) y_r + \sum _{r \in \mathbb {R}^m} \pi (r) z_r \ge 1$ for all $(y, z) \in M_f$. Given $\psi$ valid for $R_f$, we say that $\pi$ is a lifting of $\psi$ if $(\psi , \pi )$ is valid for $M_f$. For example, $(\psi , \psi )$ is a lifting of $\psi$.

Proposition 19

Let $\psi : \mathbb {R}^m \rightarrow \mathbb {R}$ be valid for $R_f$. For any $w : \mathbb {R}^m \rightarrow \mathbb {Z}^m \cap {\text {rec}}({\text {conv}}(S))$, the function $\pi (r) := \psi (r + w(r))$ is a lifting of $\psi$.

Proof

For all $(y, z) \in M_f$, we have

$$\begin{aligned} f + \sum _{r \in \mathbb {R}^m} r y_r + \sum _{r \in \mathbb {R}^m} r z_r \in S. \end{aligned}$$

Since $z_r \ge 0$, $w(r) \in \mathbb {Z}^m$, and $w(r) \in {\text {rec}}({\text {conv}}(S))$ for all $r \in \mathbb {R}^m$, we have $x + \sum _{r \in \mathbb {R}^m} w(r) z_r \in S$ for all $x \in S$. In particular,

$$\begin{aligned} f + \sum _{r \in \mathbb {R}^m} r y_r + \sum _{r \in \mathbb {R}^m} r z_r + \sum _{r \in \mathbb {R}^m} w(r) z_r \in S, \end{aligned}$$

i.e.,

$$\begin{aligned} f + \sum _{r \in \mathbb {R}^m} r y_r + \sum _{r \in \mathbb {R}^m} (r + w(r)) z_r \in S. \end{aligned}$$

Because $(\psi , \psi )$ is valid for $M_f$, we know that

$$\begin{aligned} \sum _{r \in \mathbb {R}^m} \psi (r) y_r + \sum _{r \in \mathbb {R}^m} \psi (r + w(r)) z_r = \sum _{r \in \mathbb {R}^m} \psi (r) y_r + \sum _{r \in \mathbb {R}^m} \pi (r) z_r \ge 1, \end{aligned}$$

for all $(y, z) \in M_f$. In other words, $(\psi , \pi )$ is valid for $M_f$.

Corollary 20

Let $\psi : \mathbb {R}^m \rightarrow \mathbb {R}$ be valid for $R_f$. Then,

$$\begin{aligned} \pi (r) := \min _{w \in \mathbb {Z}^m \cap {\text {rec}}({\text {conv}}(S))} \psi (r + w) \end{aligned}$$

is a lifting of $\psi$.

Proposition 19 only gives sufficient conditions for $\pi$ to be a lifting function. But if we insist on building $\pi$ with a formula of the type $\pi (r) := \psi (r + w(r))$, then in all generality, it is necessary to have $w \in \mathbb {Z}^m \cap {\text {rec}}({\text {conv}}(S))$. Proposition 21 shows that otherwise, we could construct $M_f$ such that $\pi$ is not a lifting.

Proposition 21

Let $S := \mathbb {Z}^m \cap Q$, where Q is some rational polyhedron, and $w \notin \mathbb {Z}^m \cap {\text {rec}}({\text {conv}}(S))$. There exist $f \in \mathbb {R}^m \setminus S$, $d \in \mathbb {R}^m$ and $\psi$ valid for $R_f$ such that if $\pi (d) = \psi (d + w)$, then $\pi$ is not a lifting of $\psi$.

Proof

Since $w \notin \mathbb {Z}^m \cap {\text {rec}}({\text {conv}}(S))$, there exists $\bar{x} \in S$ such that $\bar{x} + w \notin S$. Let $f := \bar{x} + w$. There exists $\varepsilon > 0$ such that $x \notin S$ for all $x \in \mathbb {R}^m$ such that $|x - f| \le \varepsilon$. Let $\psi (r) := \frac{|r|}{\varepsilon }$. It is easy to verify that $\psi$ is valid for $R_f$. We construct

$$\begin{aligned} \bar{y}_{-w} := 0, \;\; \bar{z}_{-w} := 1, \;\; \bar{y}_{t} := 0, \;\; \bar{z}_{t} := 0, \text { for all } t \ne -w. \end{aligned}$$

Clearly, $f + \sum _{r \in \mathbb {R}^m} r \bar{y}_r + \sum _{r \in \mathbb {R}^m} r \bar{z}_r = \bar{x}$ so $(\bar{y}_r, \bar{x}_r) \in M_f$. However, we can let $d := -w$ and verify that

$$\begin{aligned} \sum _{r \in \mathbb {R}^m} \psi (r) \bar{y}_r + \sum _{r \in \mathbb {R}^m} \pi (r) \bar{z}_r = \pi (-w) = \psi (0) = 0 \not \ge 1, \end{aligned}$$

showing that $(\psi , \pi )$ is not valid for $M_f$.

B Source code

As previously mentioned, the source code for the wedge cut generator presented in Sect. 6 has been made available online [30] under the GNU General Public License, version 3. In this section, we give a brief overview of its structure and usage.

For clarity and reusability, the code has been split into two modules: the module onerow/library contains the code that generates wedge cuts from a given tableau row, as well helper functions that select and extract tableau rows from CPLEX. This module compiles to a shared library, which can be used by other applications. The module onerow/benchmark, on the other hand, contains the code that measures the performance of the library on instances from the MIPLIB, as well as the scripts that convert the raw outputs into the tables presented in Sect. 6. The structure of the project is presented in Table 4. Complete instructions to compile the source code, run the computational experiments and generate the tables have been included in the file README.md.

Table 4 Directory structure and main files

Next, we describe the main classes of the library module. Given a tableau row, wedge cuts can be generated by making use of the class WedgeCutGenerator, as follows:

A Constraint is composed by a list of rational coefficients and the right-hand side. A Row is a constraint with some extra information, such as reduced costs and the index of the basic variable. Although it is possible to construct a Row manually, the library comes with a helper class CplexHelper which is able to extract it from a CPLEX model:

We recall that only rows with fractional right-hand side and integral basic variable are suitable for generating single-row cuts. The method CplexHelper::find_good_rows can be used for finding such rows. The method CplexHelper::add_cut simplifies adding the generated cut back into the CPLEX model. Before adding the cut, it verifies that the cut dynamism is within the allowed range, that it does cut off the current fractional solution and, optionally, that it does not cut off a given integral solution. Finally, instead of performing each of the previous steps in isolation, the helper class also includes the convenience method CplexHelper::add_single_row_cuts, which combines, in a single call, finding suitable rows, generating the cuts and adding them back into the model. In addition to WedgeCutGenerator, two other single-row cut generators, GomoryCutGenerator and MIRCutGenerator, have been implemented, and can be similarly used with CplexHelper. For more examples of usage, we refer to the unit tests included in the package, under onerow/library/tests.