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The mixing-MIR set with divisible capacities
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Journal Article
The mixing-MIR set with divisible capacities
2008
Overview
We study the set
, where
,
j
= 1, ...,
n
, and
B
1
| ... |
B
n
. The set
S
generalizes the mixed-integer rounding (MIR) set of Nemhauser and Wolsey and the mixing-MIR set of Günlük and Pochet. In addition, it arises as a substructure in general mixed-integer programming (MIP), such as in lot-sizing. Despite its importance, a number of basic questions about
S
remain unanswered, including the tractability of optimization over
S
and how to efficiently find a most violated cutting plane valid for
P
=
conv
(
S
). We address these questions by analyzing the extreme points and extreme rays of
P
. We give all extreme points and extreme rays of
P
. In the worst case, the number of extreme points grows exponentially with
n
. However, we show that, in some interesting cases, it is bounded by a polynomial of
n
. In such cases, it is possible to derive strong cutting planes for
P
efficiently. Finally, we use our results on the extreme points of
P
to give a polynomial-time algorithm for solving optimization over
S
.
Publisher
Springer-Verlag,Springer,Springer Nature B.V
Subject
/ Calculus of Variations and Optimal Control; Optimization
/ Exact sciences and technology
/ Inventory control, production control. Distribution
/ Mathematical and Computational Physics
/ Mathematical Methods in Physics
/ Operational research and scientific management
/ Operational research. Management science
/ Studies
