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

3.3 Algorithmic Aspects 25
resulting single mixed integer constraint and by the bounds imposed on the integer
variables is than relaxed to obtain the mixed knapsack relaxation X MK of X, as
explained in Section 3.2.
We want to comprehend the strategies suggested by Marchand and Wolsey [39,
42] for the selection of the substitutions, and we also want to develop new strategies.
Furthermore, in [39, 42], a real variable is either bounded by a simple upper bound
or by a variable upper bound. The same hold for the lower bounds: a real variable
is either bounded by a simple lower bound or by a variable lower bound. Thus, we
have to extend the bound substitution heuristic given in [39, 42] to the more general
situation considered here.
Therefore, let the MIR inequality for X MK be given in the form
j∈N
Ff ′ (α
α
0
′
j)xj +
j∈M
¯Ff ′ (γ
α
0
′
j)¯yj ≤ ⌊α ′
0⌋. (3.10)
Note that for d ≥ 0 and 0 ≤ α < 1, ¯ Fα(d) = 0. Furthermore, let k ∈ M be the index
of a real variable with γk not equal to zero in (3.9), uk < ∞, and ũk < ∞.
To understand the effect of the different bound substitutions performable for yk,
we analyze the effect on the MIR inequality (3.10) when we use one of the different
bounds imposed on yk for the substitution of yk in comparison to the case where no
bound substitution is performed.
Using a simple bound for the substitution of yk has a different effect than using a
variable bound. On the one hand, among other changes, using a simple bound may
change the value of α ′
0 and therefore, may change the value of fα ′ , which would lead
0
to different values of the coefficients of all variables in the MIR inequality (3.10).
On the other hand, among other changes, using a variable bound does not change
the value of α ′
0
, whereas the value of α′
k for the integer variable xk involved in
the variable bound used may change, which would lead to a different value of the
coefficient of xk in the MIR inequality (3.10).
In addition, using a lower bound for the substitution of yk has a different effect
than using an upper bound. On the one hand, using a lower bound does not change
the value of γ ′
k . On the other hand, using an upper bound changes the value of
γ ′
k , i.e., if γk ≥ 0 in (3.9), γ ′
k becomes negative when using an upper bound for the
substitution of yk, and the other way around. Thus, the decision of using a lower
bound or an upper bound influences the value of γ ′
k and therefore, also the coefficient
of ¯yk in the MIR inequality (3.10). If γk is nonnegative in (3.9), we assume that
in practice using a lower bound performs better than using an upper bound. If γk
is negative in (3.9), we can not decide in advance whether using a lower bound or
using an upper bound leads to a better performance.
We suggest to use a two step procedure for deciding which bound is used for the
substitution for each real variable. In the first step, we decide whether a simple or
variable bound is used, i.e., we select a lower bound lbj (simple or variable bound)
and an upper bound ubj (simple or variable). And, in the second step we decide
whether a lower or upper bound is used, i.e., we decide which of the two bounds
selected in the first step we will actually use for the substitution. From our analysis
the question arises, which of the following three criteria for the first step does lead
to the best performance of the separation algorithm in practice.