Multiple-row cuts

Let us assume to be given a mixed integer set

S = {x ∈ Rn: Ax = b, x ≥ 0, xj ∈ Z ∀j ∈ J}, (4.1) with A ∈ Qm×n _{and b ∈ Q}m_{, and denote the continuous relaxation of S as P = {x ∈} Rn_{: Ax = b, x ≥ 0}. We assume for the sake of simplicity the set S to be nonempty.}

Given a basis B ⊂ {1, . . . , n} corresponding to the vertex x∗ ∈ P \ S of the polyhe- dron P , the set S can be rewritten as

x_B = x∗ B+ P j∈Nrjxj, x ≥ 0, xj ∈ Z, j ∈ J, (4.2) where N denotes the set of nonbasic variables. A valid relaxation of S can be obtained by dropping the nonnegative restrictions on all the basic variables and considering a subset Q (with |Q| = q) of rows of (4.2) associated with basic integer-constrained variables (i.e., a subset of variables xi with i ∈ B ∩ J), thus getting

(SQ) xi = fi+ P j∈N rjxj, i ∈ Q x_j ≥ 0, j ∈ N xj ∈ Z, j ∈ J (4.3)

with fi = x∗i− bx∗ic for any i ∈ Q and fi > 0 for some i ∈ Q. In the following we assume w.l.o.g. Q = {1, . . . , q}.

Set SQ can be further relaxed by dropping all the integrality requirements on the nonbasic variables, thus getting a system of the form

x = f +Pk_j=1rj_s j, x ∈ Zq, s ∈ Rk +, (4.4)

where all the continuous variables have been renamed as s and |N | = k. Recall that f, r1_{, . . . , r}k_{∈ Q}q _{and f 6∈ Z}q_{, and denote as R}

f(r1, . . . , rk) the convex hull of all vectors

s ∈ Rk _{for which there exists x ∈ R}q _{such that (x, s) satisfies (4.4). Since S in non} empty and all the data are rational entries, R_f(r1_{, . . . , r}k_{) is a rational full-dimensional} polyhedron: i.e., it is a rational polyhedron and its recession cone is Rk

+.

Borozan and Cornu´ejols [39] considered relaxing the k-dimensional space of variables s = (s1, . . . , sk) to an infinite-dimensional space, where the variables sr are defined for any r ∈ Qq_{, thus getting the following semi-infinite relaxation, related to Gomory and} Johnson’s infinite group problem [100]:

x = f +P_r∈Qqrs_r, x ∈ Zq_,

s ≥ 0 with finite support

(4.5) where s = (sr)r∈Qq is said to have a finite support if it has a finite numbers of nonzero components. Let now R_f be the convex hull of all s ∈ RQq

for which there exists x ∈ Rq _{such that (x, s) satisfies (4.5). In this general context, they showed that any} valid inequality for Rf that cuts off the infeasible solution s = 0 can be written in the

form _X

r∈Qq

where ψ : Qq−→ Q∪{+∞} is said to be a valid function if the corresponding inequality (4.6) is valid for Rf. Then, they provided a strong correspondence between minimal valid inequalities and maximal lattice-free convex sets2 _{as follows. Let define a minimal} valid function as a valid function ψ such that there is no other valid function ψ0 with ψ0_{(r) ≤ ψ(r) for all r ∈ Q}q _{and ψ}0_{(r) < ψ(r) for some r ∈ Q}q_{. For any valid function ψ,} consider the set B_ψ = {x ∈ Qq_{: ψ(x − f ) ≤ 1} associated with ψ and denote as cl(B}

ψ) the topological closure of Bψ in Rq. The following theorem holds:

Theorem 4.1 (Borozan and Cornu´ejols [39]). Let f ∈ Qq _{\ Z}q_{. A minimal valid}

function ψ for R_f is nonnegative, piecewise linear, positively homogeneous and convex. Furthermore the set cl(B_ψ) is a full-dimensional maximal lattice-free convex set contai- ning f . Conversely, for any full-dimensional maximal lattice-free convex set B ⊂ Rq

containing f , there exists a minimal valid function ψ for R_f such that cl(B_ψ) = B, and when f is in the interior of B, this function is unique.

In practice, the above theorem shows that any minimal valid inequality for Rf arises from a maximal lattice-free convex set B containing f . Further, any maximal lattice- free convex set B containing f in the interior lead to a unique minimal valid inequality. However, the latter results does not hold if f is on the boundary of B; i.e., any maximal lattice-free convex set containing f on the boundary leads, in the general case, to several minimal valid inequalities arising from different minimal valid functions ψ. Let denote any maximal lattice-free convex set B containing f on the boundary as a degenerate set. With the same meaning, let denote any minimal valid function ψ associated with a degenerate set as a degenerate function. The following question naturally arises from Theorem 4.1: should one consider degenerate sets/functions in practice?

The answer has been provided by Zambelli [176] for the finite dimensional case (i.e., for the set R_f(r1_{, . . . , r}k_)):

Theorem 4.2 (Zambelli [176]). Given a minimal valid inequality Pk_j=1αjsj ≥ 1 for

R_f(r1_{, . . . , r}k_{), there exists a nondegenerate minimal valid function ψ such that ψ(r}j_{) =}

αj for all j = 1, . . . , k.

Thus, from the practical point of view of generating cuts for the finite dimensional set R_f(r1_{, . . . , r}k_{), one does not need to be concerned with the complications arising} from degenerate functions, since any minimal valid inequality can be obtained from a nondegenerate minimal valid function.

A first computational investigation on the possibility of generating cuts from a set of multiple rows has been provided by Espinoza [75], who considered relaxing the simplex tableau of a general MIP as a set of the form (4.4). This is obtained by selecting a subset Q of rows associated with basic integer-constrained variables and by dropping nonnegative requirements on all the basic variables and integrality requirements on all the nonbasic variables. The computational results reported in [75] on a large set of MIPLIB [36] instances showed that cutting planes from multiple-row sets are effective

in practice to improve on the lower bound at the root node yielded by the “classical” single-row cuts.