Implementation of Cutting Plane Separators for Mixed Integer ... - ZIB

2.2 Lifting Theory 7
1. If there exists λ > 0 such that (α, α0) = λ(γ, γ0), we call α T x ≤ α0 and
γ T x ≤ γ0 equivalent.
2. If there exists µ > 0 such that γ ≥ µα and γ0 ≤ µα0, then {x ∈ R n + : γ T x ≤
γ0} ⊆ {x ∈ R n + : α T x ≤ α0}. In this case, we say that γ T x ≤ γ0 is at least as
strong as α T x ≤ α0.
We assume that the MIP has an optimal solution, i.e., that the MIP is neither
infeasible nor unbounded. The general cutting plane method starts by solving the
LP relaxation of the MIP with methods from linear programming (see e.g. [52]).
Obviously, zLP ≤ zMIP holds. Let x∗ LP be an optimal solution of the LP relaxation of
the MIP. If x∗ LP is in Zp × Rn−p , we are done; x∗ LP is an optimal solution of the MIP.
Otherwise, it is a well-known result that there exists a valid inequality for X that is
from conv(X) (see [46]). Such an inequality is
not satisfied by x∗ LP , or ‘cuts off’ x∗ LP
called cutting plane and the problem of determining whether x∗ LP
is in conv(X) and
if not of finding a cutting plane is called separation problem. If we found a cutting
plane, we add it to P and obtain a polyhedron Q with conv(X) ⊆ Q ⊂ P . This
process is iterated until the solution is in Z p × R n−p , i.e., an optimal solution of the
MIP is found.
The separation problem can be formulated for different class of valid inequalities
for X. Gomory has shown that a cutting plane method based on iteratively adding
Gomory mixed integer cuts (see Chapter 6) solves a MIP under certain conditions
in a finite number of steps (see [40, 46]).
MIPs can also be solved by the linear programming based branch-and-bound algorithm.
The algorithm uses a divide-and-conquer strategy to explore the feasible
region of the MIP and therefore guarantees to find an optimal solution, if one exists.
But, instead of exploring the whole feasible region, it makes use of lower and
upper bounds and therefore avoids touching certain (large) parts of the feasible region
(see [27]). This algorithm can be used in combination with the general cutting
plane method described above. For a MIP in minimization form the cutting plane
method can help to improve the lower bound used in the branch-and-bound algorithm
(also called dual bound). The resulting algorithm is called linear programming
based branch-and-cut algorithm. We do not give a detailed description here, but refer
the interested reader to [27, 46].
2.2 Lifting Theory
For implementing efficient cutting plane separators, it seems important to use cutting
planes that are strong in the sense that they define facets of conv(X), where X is the
feasible region of the MIP, or at least faces of conv(X) of reasonably high dimension
(see [46]).
Definition 2.7 ([46]). Let P ⊆ R n be a polyhedron, and α T x ≤ α0, where α ∈ R n
and α0 ∈ R, be a valid inequality for P .
1. A set of vectors x1 , . . . , xk ∈ Rn is affinely independent if the unique solution
of k i=1 λixi = 0, k i=1 λi = 0 is λi = 0 for i = 1, . . . , k.