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1.
We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables. We develop a lifting theory for the continuous variables. In particular, we present a pseudo-polynomial algorithm for the sequential lifting of the continuous variables and we discuss its practical use.This research was supported by NSF grants DMI-0100020 and DMI-0121495Mathematics Subject Classification (2000): 90C11, 90C27 相似文献
2.
The 0-1 Knapsack problem with a single continuous variable 总被引:5,自引:0,他引:5
Specifically we investigate the polyhedral structure of the knapsack problem with a single continuous variable, called the
mixed 0-1 knapsack problem. First different classes of facet-defining inequalities are derived based on restriction and lifting. The order of
lifting, particularly of the continuous variable, plays an important role. Secondly we show that the flow cover inequalities
derived for the single node flow set, consisting of arc flows into and out of a single node with binary variable lower and
upper bounds on each arc, can be obtained from valid inequalities for the mixed 0-1 knapsack problem. Thus the separation
heuristic we derive for mixed knapsack sets can also be used to derive cuts for more general mixed 0-1 constraints. Initial
computational results on a variety of problems are presented.
Received May 22, 1997 / Revised version received December 22, 1997 Published online November 24, 1998 相似文献
3.
The supermodular covering knapsack set is the discrete upper level set of a non-decreasing supermodular function. Submodular and supermodular knapsack sets arise naturally when modeling utilities, risk and probabilistic constraints on discrete variables. In a recent paper Atamtürk and Narayanan (2009) study the lower level set of a non-decreasing submodular function.In this complementary paper we describe pack inequalities for the supermodular covering knapsack set and investigate their separation, extensions and lifting. We give sequence-independent upper bounds and lower bounds on the lifting coefficients. Furthermore, we present a computational study on using the polyhedral results derived for solving 0–1 optimization problems over conic quadratic constraints with a branch-and-cut algorithm. 相似文献
4.
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
5.
We study the set of 0–1 integer solutions to a single knapsack constraint and a set of non-overlapping cardinality constraints (MCKP), which generalizes the classical 0–1 knapsack polytope and the 0–1 knapsack polytope with generalized upper bounds. We derive strong valid inequalities for the convex hull of its feasible solutions using sequence-independent lifting. For problems with a single cardinality constraint, we derive two-dimensional superadditive lifting functions and prove that they are maximal and non-dominated under some mild conditions. We then show that these functions can be used to build strong valid inequalities for problems with multiple disjoint cardinality constraints. Finally, we present preliminary computational results aimed at evaluating the strength of the cuts obtained from sequence-independent lifting with respect to those obtained from sequential lifting. 相似文献
6.
7.
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
8.
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
9.
Mixed-integer rounding (MIR) is a simple, yet powerful procedure for generating valid inequalities for mixed-integer programs.
When used as cutting planes, MIR inequalities are very effective for mixed-integer programming problems with unbounded integer
variables. For problems with bounded integer variables, however, cutting planes based on lifting techniques appear to be more
effective. This is not surprising as lifting techniques make explicit use of the bounds on variables, whereas the MIR procedure
does not. In this paper we describe a simple procedure, which we call mingling, for incorporating variable bound information
into MIR. By explicitly using the variable bounds, the mingling procedure leads to strong inequalities for mixed-integer sets
with bounded variables. We show that facets of mixed-integer knapsack sets derived earlier by superadditive lifting techniques
can be obtained by the mingling procedure. In particular, the mingling inequalities developed in this paper subsume the continuous
cover and reverse continuous cover inequalities of Marchand and Wolsey (Math Program 85:15–33, 1999) as well as the continuous
integer knapsack cover and pack inequalities of Atamtürk (Math Program 98:145–175, 2003; Ann Oper Res 139:21–38, 2005). In
addition, mingling inequalities give a generalization of the two-step MIR inequalities of Dash and Günlük (Math Program 105:29–53,
2006) under some conditions. 相似文献
10.
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
11.
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
12.
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
13.
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
14.
Cécile Cordier Hugues Marchand Richard Laundy Laurence A. Wolsey 《Mathematical Programming》1999,86(2):335-353
A branch-and-cut mixed integer programming system, called bc–opt, is described, incorporating most of the valid inequalities that have been used or suggested for such systems, namely lifted
0-1 knapsack inequalities, 0-1 gub knapsack and integer knapsack inequalities, flowcover and continuous knapsack inequalities,
path inequalities for fixed charge network flow structure and Gomory mixed integer cuts. The principal development is a set
of interface routines allowing these cut routines to generate cuts for new subsets or aggregations of constraints.
The system is built using the XPRESS Optimisation Subroutine Library (XOSL) which includes a cut manager that handles the
tree and cut management, so that the user only essentially needs to develop the cut separation routines.
Results for the MIPLIB3.0 library are presented - 37 of the instances are solved very easily, optimal or near optimal solution
are produced for 18 other instances, and of the 4 remaining instances, 3 have 0, +1, -1 matrices for which bc–opt contains no special features.
Received May 11, 1997 / Revised version received March 8, 1999?Published online June 11, 1999 相似文献
15.
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
16.
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
17.
The Capacitated Facility Location Problem (CFLP) is to locate a set of facilities with capacity constraints, to satisfy at the minimum cost the order-demands of a set of
clients. A multi-source version of the problem is considered in which each client can be served by more than one facility.
In this paper we present a reformulation of the CFLP based on Mixed Dicut Inequalities, a family of minimum knapsack inequalities of a mixed type, containing both binary and continuous (flow) variables. By aggregating
flow variables, any Mixed Dicut Inequality turns into a binary minimum knapsack inequality with a single continuous variable.
We will refer to the convex hull of the feasible solutions of this minimum knapsack problem as the Mixed Dicut polytope.
We observe that the Mixed Dicut polytope is a rich source of valid inequalities for the CFLP: basic families of valid CFLP inequalities, like Variable Upper Bounds, Cover, Flow Cover and Effective Capacity Inequalities, are valid for the Mixed
Dicut polytope. Furthermore we observe that new families of valid inequalities for the CFLP can be derived by the lifting procedures studied for the minimum knapsack problem with a single continuous variable.
To deal with large-scale instances, we have developed a Branch-and-Cut-and-Price algorithm, where the separation algorithm
consists of the complete enumeration of the facets of the Mixed Dicut polytope for a set of candidate Mixed Dicut Inequalities.
We observe that our procedure returns inequalities that dominate most of the known classes of inequalities presented in the
literature. We report on computational experience with instances up to 1000 facilities and 1000 clients to validate the approach. 相似文献
18.
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
19.
《European Journal of Operational Research》1996,92(2):310-325
In this paper we consider the quadratic knapsack problem which consists in maximizing a positive quadratic pseudo-Boolean function subject to a linear capacity constraint. We propose a new method for computing an upper bound. This method is based on the solution of a continuous linear program constructed by adding to a classical linearization of the problem some constraints rebundant in 0–1 variables but nonredundant in continuous variables. The obtained upper bound is better than the bounds given by other known methods. We also propose an algorithm for computing a good feasible solution. This algorithm is an elaboration of the heuristic methods proposed by Chaillou, Hansen and Mahieu and by Gallo, Hammer and Simeone. The relative error between this feasible solution and the optimum solution is generally less than 1%. We show how these upper and lower bounds can be efficiently used to determine the values of some variables at the optimum. Finally we propose a branch-and-bound algorithm for solving the quadratic knapsack problem and report extensive computational tests. 相似文献
20.
Noah D. Stein Asuman Ozdaglar Pablo A. Parrilo 《International Journal of Game Theory》2008,37(4):475-504
In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form.
Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable
games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed
equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an
n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present
a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using
our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with
fixed strategy spaces in time polynomial in the rank of the game.
This research was funded in part by National Science Foundation grants DMI-0545910 and ECCS-0621922 and AFOSR MURI subaward
2003-07688-1. 相似文献