GMI cuts and disjunctive cuts are strongly related. On the one side, Nemhauser and Wolsey [128, 129] have shown that the GMI closure and the split closure relative to a polyhedron P are identical. On the other hand, Balas and Perregaard [24, 25] discovered several connections between GMIs from the tableau and basic solutions of CGLP in the context of 0–1 MIPs3.
In particular, Balas and Perregaard [25] addressed the CGLP truncated with nor- malization m X i=1 ui+ m X i=1 vi+ u0+ v0= 1
3All the results from [24, 25] were presented in the context of 0–1 MIPs. The results reported in this
and considered elementary disjunctions of the form xj ≤ 0 or xj ≥ 1. Given a basic solution x∗ of the polyhedron P and a row of the corresponding simplex tableau (lifted in the space (x, s) by including also the nonnegative surplus variables si = bi− aix) associated with a basic variable xj (j ∈ J, x∗j fractional), they first showed that the
simple disjunctive cut which can be read from the given tableau row4 corresponds to a basic (and, generally, nonoptimal) solution of the CGLP obtained by considering the elementary disjunction on variable xj. Further, they also showed that, by applying the Balas-Jeroslow [23] strengthening procedure to the above basic solution, one gets precisely the GMI cut associated with the same tableau row. Finally, they discovered a precise correspondence between the (possibly infeasible) bases of the simplex tableau of the polyhedron P in the space (x, s) and the bases of the CGLP. This allowed them to develop an elegant and efficient way of solving the CGLP by making pivot operations in the “natural” tableau involving the original x variables only (plus surplus variables). Such a method represents a crucial speed-up in the implementation of a disjunctive cut separation procedure.
Roughly speaking, the GMI cut from the tableau is a basic solution, generally no- noptimal, of CGLP. Hence, solving the CGLP, in the extended space which involves the Farkas multipliers u, v or in the original space through the procedure developed in [25], can be interpreted as a way of strengthening the GMI cut. A new investigation on this topic is presented in Chapter 3.
4Recall that the simple disjunctive cut associated with a tableau row ˜a
ix + ˜gis = x∗jis the cut which
can be obtained by applying the disjunction xj ≤ bx∗jc or xj ≥ dx∗je to the system ˜aix + ˜gis = x∗j,
Chapter 3
On the Separation of Disjunctive
Cuts
3.1
Introduction
As discussed in Chapter 1, cutting planes (and, in particular, Gomory mixed integer cuts) are probably the main ingredient behind the success of the current generation of general purpose MIP solvers. Cutting planes have been widely studied in the literature and the arsenal of separation algorithms has been continuously enlarged over the years. However, there are still several fundamental questions about the use of cutting planes which are probably not fully answered, thus reducing what we could really gain from cuts.
This chapter1presents an investigation of the main aspects related to the separation of disjunctive cuts, which, as recalled in Chapter 2, are known to be strictly related to Gomory mixed integer cuts.
In the following we consider the MIP
min{cx : Ax ≥ b, xj ∈ Z ∀j ∈ J} (3.1)
with bounds on x (if any) included in Ax ≥ b, where c ∈ Rn and A ∈ Rm×n are the given objective function and constraint matrix, while J ⊆ {1, . . . , n} denotes the set of variables constrained to be integer. For technical reasons, we assume w.l.o.g. that the system Ax ≥ b implies (or contains explicitly) the trivial inequality 0x ≥ −1, in the sense that this latter inequality can be obtained as a nonnegative combination of the rows of Ax ≥ b2.
Let x∗ denote an optimal solution of the continuous relaxation min{cx : x ∈ P }
1The results of this chapter appear in: M. Fischetti, A. Lodi and A. Tramontani, “On the separation
of disjunctive cuts”, Technical Report OR-08-2, DEIS, University of Bologna, 2008 [85].
2For problems with at least one bounded variable, the trivial inequality can always be obtained by
adding the bound constraints on a single variable, say xj≥ LBj and −xj≥ −U Bj, and dividing the
where
P := {x ∈ Rn: Ax ≥ b}. (3.2)
We are given a disjunction of the form
πx ≤ π0 OR πx ≥ π0+ 1 (3.3)
such that (π, π0) is integer, πj = 0, ∀j 6∈ J and πx∗− π0= η∗, with η∗ ∈ ]0, 1[.
In this chapter3 we are interested is deriving the “strongest” (in some sense to be di- scussed later) disjunctive cut γx ≥ γ0 violated by x∗, according to the classical approach of Balas [16]. (Disjunctive cuts which can be derived by imposing a single disjunction such as (3.3) on a polyhedron P are also known as split cuts; see Cook, Kannan and Schrijver [54].) To this end, let us denote by P0 (respectively, P1) the polyhedron ob- tained from P by imposing the additional restriction πx ≤ π0 (resp., πx ≥ π0+ 1). By Farkas lemma, the validity of γx ≥ γ0 for P0 and for P1, and hence for conv(P0∪ P1), can always be certified by means of nonnegative multipliers (u, u0, v, v0) associated with the inequalities defining P0 and P1 according to the following scheme:
P0 (u) Ax ≥ b (u0) −πx ≥ −π0 P1 (v) Ax ≥ b (v0) πx ≥ π0+ 1
A most-violated disjunctive cut can therefore be found by solving the following Cut Generating Linear Program (CGLP) that determines the Farkas multipliers so as to maximize the violation of the resulting cut with respect to the given point x∗:
(CGLP) min γx∗− γ0 (3.4) γ = uA − u0π (3.5) γ = vA + v0π (3.6) γ0 = ub − u0π0 (3.7) γ0 = vb + v0(π0+ 1). (3.8) u, v, u0, v0≥ 0 (3.9)
Note that, according to Farkas lemma, the two equations (3.7) and (3.8) defining γ0 should be relaxed into ≤ inequalities. However it is not difficult to see that, due to the (possibly implicit) presence of the trivial inequality 0x ≥ −1, one can always require that equality holds in both cases.
By construction, any feasible CGLP solution with negative objective function value corresponds to a violated disjunctive cut. However, as stated, the feasible CGLP set is
3The main steps related to a typical disjunctive cut separation procedure (i.e., projection onto the
support, subsequent lifting, a posteriori cut strengthening by changing the disjunction) have been already discussed in Chapter 2. To avoid confusion, these steps are briefly recalled even in this chapter, in the context of the required different notation, in which the bounds on x variables are eventually included in the system Ax ≥ b.
a cone and needs to be truncated so as to produce a bounded LP in case a violated cut exists. This crucial step will be addressed in the next section.
Usually, the CGLP is projected onto the support of x∗. Given a variable x
krestricted to be nonnegative and such that x∗
k= 04, it is well known [20] that one can project xk away. More precisely, one can avoid considering the CGLP constraints associated with γk and neglect constraint xk≥ 0 in both P0 and P1. The resulting (reduced) CGLP is then solved and the cut coefficient γk is derived afterwards by solving the trivial lifting problem
min{γk: γk= uAk− u0πk = vAk+ v0πk, u, v ≥ 0}, (3.10) where the Farkas multipliers u and v are fixed as in the optimal solution of the reduced CGLP but those related to the previously neglected bound constraint xk≥ 0.
In practice, disjunction (3.3) is typically elementary, i.e., it involves only one integer variable and it reads xj ≤ bx∗
jc OR xj ≥ dx∗je, with j ∈ J and x∗j fractional. As such, the disjunctive cut only exploits the integrality requirement on a single variable and can therefore be improved easily by an a posteriori cut strengthening procedure as the one proposed by Balas and Jeroslow [23]. As already discussed in Chapter 2, such a strengthening can be also interpreted as finding the best disjunction for the given set of multipliers.
Recently, Balas and Perregaard [25] developed an elegant and efficient way of solving the CGLP by making pivot operations in the “natural” tableau involving the original x variables only (plus surplus variables), which represents a crucial speed-up in the implementation of the method.
In this chapter we investigate computationally the main ingredients of a disjunctive cut separation procedure, and analyze their impact on the overall performance at the root node of the branching tree. To be more specific, we consider a testbed of MIPs taken from MIPLIB library [36]. For each instance, we solve the root-node LP rela- xation and generate 10 rounds of disjunctive cuts computed according to alternative strategies. In each round, a violated disjunctive cut is generated for each fractional LP components x∗
j, by exploiting the disjunction xj ≤ bx∗jc OR xj ≥ dx∗je. In order to limit possible side effects, no a posteriori cut strengthening procedure is applied, unless otherwise stated.
The chapter is organized as follows. In Section 3.2 we compare classical normali- zation conditions used to truncate the CGLP cone, and try to better understand their role. In Section 3.3 we characterize weak rays/vertices of the CGLP leading to domi- nated cuts and we propose a practical heuristic method to strengthen them. In Section 3.4 we show that using redundant constraints in the CGLP can lead to very weak cuts, and we analyze such an issue with respect to the normalization used. In Section 3.5 we introduce a new normalization which is particularly suited for set-covering type pro- blems and we analyze its theoretical properties and computational behavior. Finally, some conclusions are drawn in Section 3.6.
4Of course, variables with nonzero lower bound can be shifted, while variables at the upper bound