For any given disjunction of the form
πx ≤ π0 OR πx ≥ π0+ 1 (2.2)
such that (π, π0) ∈ Zn+1, π
j = 0 ∀j 6∈ J, let P0 (respectively, P1) be the polyhedron obtained from P by imposing the additional restriction πx ≤ π0 (resp., πx ≥ π0+ 1). A disjunctive inequality is an inequality γx ≥ γ0 valid both for P0 and P1, and then also for conv(P0∪ P1) and hence for conv(S). Disjunctive cuts which can be derived by imposing a single disjunction such as (2.2) on a polyhedron P are also known as split cuts; see Cook, Kannan, and Schrijver [54]. The intersection of all split inequalities (for all the possible disjunctions of the form (2.2)) is called the split closure relative to P .
By Farkas lemma, the validity of γx ≥ γ0 for P0and for P1can 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
Given a fractional solution x∗ of P , a most-violated disjunctive cut can therefore be found by solving the following problem that determines the disjunction (π, π0) and the Farkas multipliers so as to maximize the violation of the resulting cut with respect to the given point x∗:
min γx∗− γ 0 γ ≥ uA − u0π γ ≥ vA + v0π γ0 ≤ ub − u0π0 γ0 ≤ vb + v0(π0+ 1) u, v, u0, v0≥ 0 π0∈ Z πj ∈ Z, j ∈ J πj = 0, j 6∈ J. (2.3)
By construction, any feasible solution with negative objective function value in (2.3) corresponds to a violated disjunctive cut. However, when solving the separation pro- blem (2.3), there are two different aspects to be considered. On the one side, (2.3) is a mixed integer nonlinear program involving products of integer and continuous variables. On the other hand, even if the disjunction is fixed a priori, the resulting Linear Program (LP) is defined on a cone and needs to be truncated so as to produce a bounded LP in case a violated cut exists. From a theoretical point of view, Caprara and Letchford [42] have formulated the problem of optimizing over the split closure as a mixed integer nonlinear program and have shown that the separation problem for split cuts is stron- gly N P-hard. On a practical side, Balas and Saxena [26] addressed the problem by exploiting the normalization condition u0 + v0 = 1, which allows one to restate (2.3) and to solve it as a parametric MIP with a single parameter. A similar approach was also proposed by Dash, G¨unl¨uk and Lodi [66, 67] in the context of the Mixed Integer Rounding (MIR) closure1.
A typical approach for separating disjunctive cuts in a branch-and-cut framework is the one arising from Balas, Ceria and Cornu´ejols [20], which showed the effectiveness of lift-and-project cuts in the context of 0–1 MIPs2. This approach can be easily extended to general MIPs (i.e., MIPs with general integer-constrained variables) and can be de- scribed as follows. Given the current fractional solution x∗ of the current LP relaxation, a violated disjunction of the form (2.2) is fixed, with πx∗− π
0 ∈ ]0, 1[, and a disjunctive cut is separated by solving the so-called Cut Generating Linear Program (CGLP):
(CGLP) min γx∗− γ 0 γ ≥ uA − u0π γ ≥ vA + v0π γ0≤ ub − u0π0 γ0≤ vb + v0(π0+ 1) u, v, u0, v0 ≥ 0. (2.4)
1Note that the MIR closure and the Split closure define the same polyhedron; see e.g., [57, 66, 67]. 2Lift-and-project cuts are a particular class of disjunctive cuts for 0–1 MIPs arising from elementary
As already stated, CGLP is defined on a cone and needs to be truncated with a sui- table normalization condition so as to produce a bounded LP in case a violated cut exists. Different normalization conditions lead in general to very different results. The connections among normalization conditions, cut strength and other related issues are investigated in Chapter 3 of this thesis.
Usually, the CGLP is projected onto the support of x∗ (see, e.g., [20]). Given a variable xk such that x∗k = 0, the value of the cut coefficient γk does not affect the cut violation. Hence, one can avoid considering the CGLP constraints associated with γk. The resulting (reduced) CGLP is then solved and the cut coefficient γk is derived afterwards as
γk= max{uAk− u0πk, vAk+ v0πk}, (2.5) where all the Farkas multipliers are fixed as in the optimal solution of the reduced CGLP.
In practice, the disjunction selected for solving CGLP is typically elementary, i.e., it involves only one integer variable, thus reading xj ≤ bx∗jc OR xj ≤ dx∗je (j ∈ J). As such, the disjunctive cut only exploits the integrality requirement on a single variable and can therefore be easily improved by an a posteriori cut strengthening procedure as the one proposed by Balas and Jeroslow [23].
Theorem 2.1 (Balas and Jeroslow [23]). Let (γ, γ0, u, v, u0, v0) be an optimal solution of CGLP with respect to a certain disjunction (π, π0). Define mj = uAuj0−vA+v0j, and
˜ γj =
½
min{uAj − u0bmjc, vAj+ v0dmje} if j ∈ J,
max{uAj, vAj} if j 6∈ J.
Then the inequality ˜γx ≥ γ0 is valid for conv(S) and dominates γx ≥ γ0 over the set {x ∈ Rn: x ≥ 0}.
The strengthening described in the above theorem can be interpreted as finding the best disjunction for the given set of multipliers, by fixing πj = bmjc or πj = dmje for any j ∈ J.