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

3.4 Computational Study 37
is also the case for substituting the integer variables in the cut generation heuristic.
Thus, in our resulting algorithm (slow version) we use the same criteria for selecting
the bounds for the substitution as in the default algorithm, i.e., Criterium F3 and
Criterium S3.
We also tested the modification to perform the bound substitution heuristic
in addition to the current aggregated constraint multiplied by minus one, which
Gonçalves and Ladanyi suggested in [29]. The results for our main test set are given
in Table B.10 and a summary of the results is contained in Table 3.1. For some of
the instances in our main test set, the modification leads to a greater value of the
initial gap closed while for others it leads to a smaller value. The initial gap closed
in geometric mean reduces by 1.19 percentage points. In addition, the time spent in
the separation algorithm in total increases by 5642.8 seconds of CPU time. Thus, we
have decided not to use this modification in our resulting algorithm (slow version).
Cut Generation Heuristic
In Section 3.3.3, we introduced the cut generation heuristic suggested by Marchand
and Wolsey [39, 42], which we denoted by Procedure 1. This procedure is used in
our default algorithm.
We tested it against Procedure 2 and Procedure 3, which differ from Procedure 1
by the chosen initial partition (T, U) of N. The results for our main test set are
given in Table B.11 and Table B.12 (see also Table 3.1, for a summary of the results).
Neither of them leads to an improved performance of the separation algorithm.
The initial gap closed in geometric mean reduces by 3.28 percentage points when
using Procedure 2, i.e., when complementing initially all integer variables xj, j ∈
N with a negative coefficient in the constructed mixed knapsack set, and by 0.64
percentage points when using Procedure 3, i.e., when complementing initially all
integer variables xj, j ∈ N with a nonnegative coefficient in the constructed mixed
knapsack set. We conclude that, analogue to the bound substitution heuristic, in
practice complementing all integer variables xj, j ∈ N for which x ∗ j
is closer to
the upper bound than to the lower bound leads to the best performance of the
separation algorithm. Thus, in our resulting algorithm (slow version) we use the
same procedure for the cut generation heuristic as in the default algorithm, i.e., we
use Procedure 1.
In addition, we tested to use the extended candidate set for the value of δ. The
results for our main test set given in Table B.13 show that this modification improves
the performance of the separation algorithm (see also Table 3.1, for a summary of
the results). The initial gap closed in geometric mean increases by 0.93 percentage
points. Thus, we will use the extended candidate set for the value of δ in our
resulting algorithm (slow version).
In Procedure 1, we complement additional integer variables if fβ = 0 for all δ ∈
N ∗ . Since this step is not mentioned in [42] and [29], we have investigated the effect
of this step on the performance of the separation algorithm, by testing Procedure 1
without this step. The results for our main test set are given in Table B.14 and a
summary of the results is contained in Table 3.1. The initial gap closed in geometric
mean reduces by 0.73 percentage points. We conclude that the step is useful for
an efficient cutting plane separator and will therefore not leave it in our resulting