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

44 Chapter 4. Cutting Plane Separator for the 0-1 Knapsack Problem
33, 12, 51, 58]. This procedure involves the solution of a knapsack problem in every
lifting step. Zemel [61] has introduced a polynomial algorithm which uses dynamic
programming to solve these knapsack problems. Another type of lifting is called
down-lifting, where variables fixed at their upper bounds are lifted. The idea of
sequential down-lifting has been introduced in [59], where it was used to strengthen
canonical inequalities.
The inequalities derived by sequential lifting, in general, depend on the sequence
in which the variables are lifted. A simultaneous up-lifting procedure to strengthen
minimal cover inequalities was studied by Balas and Zemel [12], but its computational
burden prevents it from being applied in practice. In [60], the property of
superadditivity of the lifting function has been explored which leads to sequence
independent lifting. Building on the results of [60], Gu, Nemhauser and Savelsbergh
[32] and Atamturk [5] have investigated the concept of superadditive up-lifting to
strengthen minimal cover inequalities. Here, the lifting function which, in general, is
not superadditive is approximated by a so-called superadditive valid lifting function
to obtain sequence independent lifting (see Section 2.2).
The results of the theoretical study of the 0-1 knapsack polytope have been used
in linear programming based branch-and-cut algorithms to solve IPs and MIPs.
Crowder, Johnson and Padberg [22] pioneered the use of lifted inequalities and
successfully solved several instances of IPs which were, at the time, considered to
be unsolvable. They separated the class of lifted minimal cover inequalities using
sequential up-lifting and the class of lifted (1,k)-configuration inequalities using sequential
up-lifting. Since then, there have been several other successful applications
of lifted valid inequalities for the 0-1 knapsack polytope. Van Roy and Wolsey [55]
separated the class of lifted cover inequalities using sequential up- and down-lifting.
Hoffman and Padberg [35] and Gu, Nemhauser and Savelsbergh [30] implemented
a cutting plane separator which separates the class of lifted minimal cover inequalities
using sequential up- and down-lifting (LMCI1). In [30], a computational study
was presented in which many of the algorithmic and implementation choices were
evaluated which have to be made when implementing this cutting plane separator.
Especially, it turned out that using both up-lifting and down-lifting instead of using
only up-lifting leads to a better performance of the cutting plane separator. Martin
[44] separated the class of lifted extended weight inequalities using sequential upand
down-lifting (LEWI).
For our cutting plane separator, we have to decide which of the above mentioned
classes of valid inequalities we want to separate. To our knowledge, no paper has
been published presenting computational results for separating the class of lifted
minimal cover inequalities using superadditive up-lifting (LMCI2). Besides, we know
of no paper in which the performance of a separation algorithm for the class of
LMCI1 has been compared to that of a separation algorithm for the class of LEWI.
Thus, it is not clear, which of these three classes of valid inequalities leads to a best
performing cutting plane separator. Therefore, we investigate separation algorithms
for all three classes of valid inequalities in this chapter.
In Section 4.2, we give a brief introduction to these classes of valid inequalities,
and in Section 4.3, we discuss different algorithmic aspects of the corresponding separation
algorithms. In Section 4.4, we present a computational study. It evaluates
the effect of using the different algorithmic and implementation choices described