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1.
The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi (Math Program 120(2):313–346, 2009) are valid inequalities for the mixed-integer knapsack set that are derived by using periodic n-step MIR functions and define facets for group problems. The mingling and 2-step mingling inequalities of Atamtürk and Günlük (Math Program 123(2):315–338, 2010) are also derived based on MIR and they incorporate upper bounds on the integer variables. It has been shown that these inequalities are facet-defining for the mixed integer knapsack set under certain conditions and generalize several well-known valid inequalities. In this paper, we introduce new classes of valid inequalities for the mixed-integer knapsack set with bounded integer variables, which we call n-step mingling inequalities (for positive integer n). These inequalities incorporate upper bounds on integer variables into n-step MIR and, therefore, unify the concepts of n-step MIR and mingling in a single class of inequalities. Furthermore, we show that for each n, the n-step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions. For n?=?2, we extend the results of Atamtürk and Günlük on facet-defining properties of 2-step mingling inequalities. For n ≥ 3, we present new facets for the mixed integer knapsack set. As a special case we also derive conditions under which the n-step MIR inequalities define facets for the mixed integer knapsack set. In addition, we show that n-step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets. 相似文献
2.
In this survey we attempt to give a unified presentation of a variety of results on the lifting of valid inequalities, as
well as a standard procedure combining mixed integer rounding with lifting for the development of strong valid inequalities
for knapsack and single node flow sets. Our hope is that the latter can be used in practice to generate cutting planes for
mixed integer programs.
The survey contains essentially two parts. In the first we present lifting in a very general way, emphasizing superadditive
lifting which allows one to lift simultaneously different sets of variables. In the second, our procedure for generating strong
valid inequalities consists of reduction to a knapsack set with a single continuous variable, construction of a mixed integer
rounding inequality, and superadditive lifting. It is applied to several generalizations of the 0–1 single node flow set.
This paper appeared in 4OR, 1, 173–208 (2003).
The first author is supported by the FNRS as a chercheur qualifié. This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction
initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed
by the authors. 相似文献
3.
Mixed-integer rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixed-integer
programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities.?Given
a mixed-integer region S and a collection of valid “base” mixed-integer inequalities, we develop a procedure for generating new valid inequalities
for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset
of these MIR inequalities, we generate two new inequalities by combining or “mixing” them. We show that the new inequalities
are strong in the sense that they fully describe the convex hull of a special mixed-integer region associated with the base
inequalities.?We discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various
mixed-integer programming problems. In particular, we present examples for production planning, capacitated facility location,
capacitated network design, and multiple knapsack problems. We also present preliminary computational results using the mixing
procedure to tighten the formulation of some difficult integer programs. Finally we study some extensions of this mixing procedure.
Received: April 1998 / Accepted: January 2001?Published online April 12, 2001 相似文献
4.
A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative
variables are allowed to be positive. This structure occurs, for example, in areas such as finance, location, and scheduling.
Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that
relate the continuous and the 0-1 variables. We use an alternative approach, in which we keep in the model only the continuous
variables, and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities
valid for the convex hull of the feasible set in the space of the continuous variables. To derive the valid inequalities,
we extend the concepts of cover and cover inequality, commonly used in 0-1 programming, to this class of problems, and we
show how cover inequalities can be lifted to derive facet-defining inequalities. We present three families of non-trivial
facet-defining inequalities that are lifted cover inequalities. Finally, we report computational results that demonstrate
the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0-1 variables
over the traditional MIP approach for this class of problems.
Received: March 13, 2003
Published online: April 10, 2003
Key Words. mixed-integer programming – knapsack problem – cardinality constrained programming – branch-and-cut 相似文献
5.
Santanu S. Dey Jean-Philippe P. Richard Yanjun Li Lisa A. Miller 《Mathematical Programming》2010,121(1):145-170
Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to
generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate
infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important
since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties
of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities
are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new
properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of
a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group
problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite
group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res
Lett 30(2):74–82, 2002), Dash and Günlük (Proceedings 10th conference on integer programming and combinatorial optimization.
Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We
also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining
extreme inequalities for infinite group relaxations of mixed integer programs are continuous.
S.S. Dey and J.-P.P. Richard was supported by NSF Grant DMI-03-48611. 相似文献
6.
We introduce a new class of second-order
cover inequalities whose members are generally stronger than the classical knapsack cover inequalities that are commonly used to enhance the
performance of branch-and-cut methods for 0–1 integer programming problems. These inequalities result by focusing attention
on a single knapsack constraint in addition to an inequality that bounds the sum of all variables, or in general, that bounds
a linear form containing only the coefficients 0, 1, and –1. We provide an algorithm that generates all non-dominated second-order
cover inequalities, making use of theorems on dominance relationships to bypass the examination of many dominated alternatives.
Furthermore, we derive conditions under which these non-dominated second-order cover inequalities would be facets of the convex
hull of feasible solutions to the parent constraints, and demonstrate how they can be lifted otherwise. Numerical examples
of applying the algorithm disclose its ability to generate valid inequalities that are sometimes significantly stronger than
those derived from traditional knapsack covers. Our results can also be extended to incorporate multiple choice inequalities
that limit sums over disjoint subsets of variables to be at most one.
相似文献
7.
Andrzej Ruszczyński 《Mathematical Programming》2002,93(2):195-215
We consider stochastic programming problems with probabilistic constraints involving random variables with discrete distributions.
They can be reformulated as large scale mixed integer programming problems with knapsack constraints. Using specific properties
of stochastic programming problems and bounds on the probability of the union of events we develop new valid inequalities
for these mixed integer programming problems. We also develop methods for lifting these inequalities. These procedures are
used in a general iterative algorithm for solving probabilistically constrained problems. The results are illustrated with
a numerical example.
Received: October 8, 2000 / Accepted: August 13, 2002 Published online: September 27, 2002
Key words. stochastic programming – integer programming – valid inequalities 相似文献
8.
Ricardo Fukasawa Laurent Poirrier Álinson S. Xavier 《Mathematical Programming Computation》2018,10(3):423-455
We consider the problem of generating inequalities that are valid for one-row relaxations of a simplex tableau, with the integrality constraints preserved for one or more non-basic variables. These relaxations are interesting because they can be used to generate cutting planes for general mixed-integer problems. We first consider the case of a single non-basic integer variable. This relaxation is related to a simple knapsack set with two integer variables and two continuous variables. We study its facial structure by rewriting it as a constrained two-row model, and prove that all its facets arise from a finite number of maximal \(\left( \mathbb {Z}\times \mathbb {Z}_+\right) \)-free splits and wedges. The resulting cuts generalize both MIR and 2-step MIR inequalities. Then, we describe an algorithm for enumerating all the maximal \(\left( \mathbb {Z}\times \mathbb {Z}_+\right) \)-free sets corresponding to facet-defining inequalities, and we provide an upper bound on the split rank of those inequalities. Finally, we run computational experiments to compare the strength of wedge cuts against MIR cuts. In our computations, we use the so-called trivial fill-in function to exploit the integrality of more non-basic variables. To that end, we present a practical algorithm for computing the coefficients of this lifting function. 相似文献
9.
A dynamic knapsack set is a natural generalization of the 0-1 knapsack set with a continuous variable studied recently. For
dynamic knapsack sets a large family of facet-defining inequalities, called dynamic knapsack inequalities, are derived by
fixing variables to one and then lifting. Surprisingly such inequalities have the simultaneous lifting property, and for small
instances provide a significant proportion of all the facet-defining inequalities.
We then consider single-item capacitated lot-sizing problems, and propose the joint study of three related sets. The first
models the discrete lot-sizing problem, the second the continuous lot-sizing problem with Wagner-Whitin costs, and the third
the continuous lot-sizing problem with arbitrary costs. The first set that arises is precisely a dynamic knapsack set, the
second an intersection of dynamic knapsack sets, and the unrestricted problem can be viewed as both a relaxation and a restriction
of the second. It follows that the dynamic knapsack inequalities and their generalizations provide strong valid inequalities
for all three sets.
Some limited computation results are reported as an initial test of the effectiveness of these inequalities on capacitated
lot-sizing problems.
Received: March 28, 2001 / Accepted: November 9, 2001 Published online: September 27, 2002
RID="★"
ID="★" Research carried out with financial support of the project TMR-DONET nr. ERB FMRX–CT98–0202 of the European Union.
Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium.
Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium.
Key words. knapsack sets – valid inequalities – simultaneous lifting – lot-sizing – Wagner-Whitin costs 相似文献
10.
We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables, through the lifting of continuous variables fixed at their upper bounds. We introduce the concept of a superlinear inequality and show that, in this case, lifting is significantly simpler than for general inequalities. We use the superlinearity theory, together with the traditional lifting of 0-1 variables, to describe families of facets of the mixed 0-1 knapsack polytope. Finally, we show that superlinearity results can be extended to nonsuperlinear inequalities when the coefficients of the variables fixed at their upper bounds are large.This research was supported by NSF grants DMI-0100020 and DMI-0121495Mathematics Subject Classification (1991): 90C11, 90C27 相似文献
11.
We present a generalization of the mixed integer rounding (MIR) approach for generating valid inequalities for (mixed) integer
programming (MIP) problems. For any positive integer n, we develop n facets for a certain (n + 1)-dimensional single-constraint polyhedron in a sequential manner. We then show that for any n, the last of these facets (which we call the n-step MIR facet) can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, which we
refer to as the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and günlük (Math Program 105(1):29–53, 2006) are the
first two families corresponding to n = 1,2, respectively. The n-step MIR inequalities are easily produced using periodic functions which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Finally, we prove that the n-step MIR inequalities generate two-slope facets for the infinite group polyhedra, and hence are potentially strong.
相似文献
12.
In this paper, we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs.
The theory is geometric and provides insights into lifting-based cut generation procedures, yielding short proofs of earlier
results in mixed-integer programming. Using convex extensions, we obtain conditions that allow for sequence-independent lifting
in nonlinear settings, paving a way for efficient cut-generation procedures for nonlinear programs. This sequence-independent
lifting framework also subsumes the superadditive lifting theory that has been used to generate many general-purpose, strong
cuts for integer programs. We specialize our lifting results to derive facet-defining inequalities for mixed-integer bilinear
knapsack sets. Finally, we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained
using a single round of traditional integer programming cut-generation techniques applied on a tight reformulation of the
problem.
相似文献
13.
Alper Atamtürk 《Mathematical Programming》2003,98(1-3):145-175
We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed–integer programming.Supported, in part, by NSF grants DMII–0070127 and DMII–0218265.Mathematics Subject Classification (2000): 90C10, 90C11, 90C57 相似文献
14.
We study the interpolation procedure of Gomory and Johnson (1972), which generates cutting planes for general integer programs
from facets of cyclic group polyhedra. This idea has recently been re-considered by Evans (2002) and Gomory, Johnson and Evans
(2003). We compare inequalities generated by this procedure with mixed-integer rounding (MIR) based inequalities discussed
in Dash and Gunluk (2003). We first analyze and extend the shooting experiment described in Gomory, Johnson and Evans. We
show that MIR based inequalities dominate inequalities generated by the interpolation procedure in some important cases. We
also show that the Gomory mixed-integer cut is likely to dominate any inequality generated by the interpolation procedure
in a certain probabilistic sense. We also generalize a result of Cornuéjols, Li and Vandenbussche (2003) on comparing the
strength of the Gomory mixed-integer cut with related inequalities. 相似文献
15.
In this survey we attempt to give a unified presentation of a variety of results on the lifting of valid inequalities, as well as a standard procedure combining mixed integer rounding with lifting for the development of strong valid inequalities for knapsack and single node flow sets. Our hope is that the latter can be used in practice to generate cutting planes for mixed integer programs. The survey contains essentially two parts. In the first we present lifting in a very general way, emphasizing superadditive lifting which allows one to lift simultaneously different sets of variables. In the second, our procedure for generating strong valid inequalities consists of reduction to a knapsack set with a single continuous variable, construction of a mixed integer rounding inequality, and superadditive lifting. It is applied to several generalizations of the 0-1 single node flow set.Received: December 2002, Revised: April 2003, AMS classification:
90C11, 90C27Laurence A. Wolsey: Corresponding author: CORE, Voie du Roman Pays 34, 1348 Louvain-la-Neuve, Belgium. The first author is supported by the FNRS as a research fellow. This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Ministers Office, Science Policy Programming. The scientific responsibility is assumed by the authors.Laurence A. Wolsey: This research was also supported by the European Commission GROWTH Programme, Research Project LISCOS, Large Scale Integrated Supply Chain Optimization Software Based on Branch-and-Cut and Constraint Programming Methods, Contract No. GRDI-1999-10056, and the project TMR-DONET nr. ERB FMRX-CT98-0202. 相似文献
16.
In this paper we discuss the derivation of strong valid inequalities for (mixed) integer knapsack sets based on lifting of
valid inequalities for basic knapsack sets with two integer variables (and one continuous variable). The basic polyhedra can
be described in polynomial time. We use superadditive valid lifting functions in order to obtain sequence independent lifting.
Most of these superadditive functions and valid inequalities are not obtained in polynomial time. 相似文献
17.
Andrew J. Miller George L. Nemhauser Martin W.P. Savelsbergh 《Mathematical Programming》2003,94(2-3):375-405
We present and study a mixed integer programming model that arises as a substructure in many industrial applications. This
model generalizes a number of structured MIP models previously studied, and it provides a relaxation of various capacitated
production planning problems and other fixed charge network flow problems. We analyze the polyhedral structure of the convex
hull of this model, as well as of a strengthened LP relaxation. Among other results, we present valid inequalities that induce
facets of the convex hull under certain conditions. We also discuss how to strengthen these inequalities by using known results
for lifting valid inequalities for 0–1 continuous knapsack problems.
Received: 30 October 2000 / Accepted: 25 March 2002 Published online: September 27, 2002
Key words. mixed integer programming – production planning – polyhedral combinatorics – capacitated lot–sizing – fixed charge network
flow
Some of the results of this paper have appeared in condensed form in ``Facets, algorithms, and polyhedral characterizations
of a multi-item production planning model with setup times', Proceedings of the Eighth Annual IPCO conference, pp. 318-332, by the same authors.
This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian
State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors.
This research was also supported by NSF Grant No. DMI-9700285 and by Philips Electronics North America. 相似文献
18.
Cover inequalities are commonly used cutting planes for the 0–1 knapsack problem. This paper describes a linear-time algorithm (assuming the knapsack is sorted) to simultaneously lift a set of variables into a cover inequality. Conditions for this process to result in valid and facet-defining inequalities are presented. In many instances, the resulting simultaneously lifted cover inequality cannot be obtained by sequentially lifting over any cover inequality. Some computational results demonstrate that simultaneously lifted cover inequalities are plentiful, easy to find and can be computationally beneficial. 相似文献
19.
Sebastián Ceria Cécile Cordier Hugues Marchand Laurence A. Wolsey 《Mathematical Programming》1998,81(2):201-214
We investigate the use of cutting planes for integer programs with general integer variables. We show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0–1 variables. We also explore the use of Gomory's mixed-integer cuts. We address both theoretical and computational issues and show how to embed these cutting planes in a branch-and-bound framework. We compare results obtained by using our cut generation routines in two existing systems with a commercially available branch-and-bound code on a range of test problems arising from practical applications. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.This research was partly performed when the author was affiliated with CORE, Université Catholique de Louvain. 相似文献
20.
《Operations Research Letters》2020,48(5):607-611
Valid inequalities for the knapsack polytope have proven to be very useful in exact algorithms for mixed-integer linear programming. In this paper, we focus on the knapsack cover inequalities, introduced in 2000 by Carr and co-authors. In general, these inequalities can be rather weak. To strengthen them, we use lifting. Since exact lifting can be time-consuming, we present two fast approximate lifting procedures. The first procedure is based on mixed-integer rounding, whereas the second uses superadditivity. 相似文献