Simultaneously lifting sets of binary variables into cover inequalities for knapsack polytopes

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.     

Lifted inequalities for 0\mathord {-}1 mixed-integer bilinear covering sets

In this paper, we study $0\mathord {-}1$ mixed-integer bilinear covering sets. We derive several families of facet-defining inequalities via sequence-independent lifting techniques. We then show that these sets have a polyhedral structure that is similar to that of a certain fixed-charge single-node flow set. As a result, we also obtain new facet-defining inequalities for the single-node flow set that generalize well-known lifted flow cover inequalities from the integer programming literature.     

Lifting inequalities: a framework for generating strong cuts for nonlinear programs

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.      

Sequence Independent Lifting for Mixed Integer Programs with Variable Upper Bounds

We investigate the convex hull of the set defined by a single inequality with continuous and binary variables with variable upper bound constraints. We extend the traditional flow cover inequality, and show that it is valid for a restriction of the set in which some variables are fixed. We also give conditions under which this inequality is facet-defining and, when it is not, we show how it can be lifted to obtain valid inequalities for the entire set using sequence independent lifting. In general, computing the lifting function is NP-hard, but under an additional restriction on the cover we obtain a closed form. Finally, we show how these results imply and extend known results about the single node fixed charge flow polyhedron. This material is based upon work supported by the National Science Foundation under Grant No. 0084826. Received: April 2004     

The 0-1 Knapsack problem with a single continuous variable

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     

Lifted flow cover inequalities for mixed 0-1 integer programs

We investigate strong inequalities for mixed 0-1 integer programs derived from flow cover inequalities. Flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities. However, because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients, lifting of flow cover inequalities is computationally very demanding. We present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and give computational results that show the effectiveness of our lifting procedures. Received May 15, 1996 / Revised version received August 7, 1998 Published online June 28, 1999     

The graphical relaxation: A new framework for the symmetric traveling salesman polytope

A present trend in the study of theSymmetric Traveling Salesman Polytope (STSP(n)) is to use, as a relaxation of the polytope, thegraphical relaxation (GTSP(n)) rather than the traditionalmonotone relaxation which seems to have attained its limits. In this paper, we show the very close relationship between STSP(n) and GTSP(n). In particular, we prove that every non-trivial facet of STSP(n) is the intersection ofn + 1 facets of GTSP(n),n of which are defined by the degree inequalities. This fact permits us to define a standard form for the facet-defining inequalities for STSP(n), that we calltight triangular, and to devise a proof technique that can be used to show that many known facet-defining inequalities for GTSP(n) define also facets of STSP(n). In addition, we give conditions that permit to obtain facet-defining inequalities by composition of facet-defining inequalities for STSP(n) and general lifting theorems to derive facet-defining inequalities for STSP(n +k) from inequalities defining facets of STSP(n).Partially financed by P.R.C. Mathématique et Informatique.     

Lifting valid inequalities for the precedence constrained knapsack problem

This paper considers the precedence constrained knapsack problem. More specifically, we are interested in classes of valid inequalities which are facet-defining for the precedence constrained knapsack polytope. We study the complexity of obtaining these facets using the standard sequential lifting procedure. Applying this procedure requires solving a combinatorial problem. For valid inequalities arising from minimal induced covers, we identify a class of lifting coefficients for which this problem can be solved in polynomial time, by using a supermodular function, and for which the values of the lifting coefficients have a combinatorial interpretation. For the remaining lifting coefficients it is shown that this optimization problem is strongly NP-hard. The same lifting procedure can be applied to (1,k)-configurations, although in this case, the same combinatorial interpretation no longer applies. We also consider K-covers, to which the same procedure need not apply in general. We show that facets of the polytope can still be generated using a similar lifting technique. For tree knapsack problems, we observe that all lifting coefficients can be obtained in polynomial time. Computational experiments indicate that these facets significantly strengthen the LP-relaxation. Received July 10, 1997 / Revised version received January 9, 1999? Published online May 12, 1999     

A polyhedral study of the cardinality constrained knapsack problem

 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     

Dynamic knapsack sets and capacitated lot-sizing

 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     

Lifting the knapsack cover inequalities for the knapsack polytope

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.     

On a generalization of the master cyclic group polyhedron

We study the master equality polyhedron (MEP) which generalizes the master cyclic group polyhedron (MCGP) and the master knapsack polyhedron (MKP). We present an explicit characterization of the polar of the nontrivial facet-defining inequalities for MEP. This result generalizes similar results for the MCGP by Gomory (1969) and for the MKP by Araóz (1974). Furthermore, this characterization gives a polynomial time algorithm for separating an arbitrary point from MEP. We describe how facet-defining inequalities for the MCGP can be lifted to obtain facet-defining inequalities for MEP, and also present facet-defining inequalities for MEP that cannot be obtained in such a way. Finally, we study the mixed-integer extension of MEP and present an interpolation theorem that produces valid inequalities for general mixed integer programming problems using facets of MEP.     

Generalised 2-circulant inequalities for the max-cut problem

The max-cut problem is a fundamental combinatorial optimisation problem, with many applications. Poljak and Turzik found some facet-defining inequalities for the associated polytope, which we call 2-circulant inequalities. We present a more general family of facet-defining inequalities, an exact separation algorithm that runs in polynomial time, and some computational results.     

Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs

This paper presents a new class of valid inequalities for the single-item capacitated lot sizing problem with step-wise production costs (LS-SW). Constant sized batch production is carried out with a limited production capacity in order to satisfy the customer demand over a finite horizon. A new class of valid inequalities we call mixed flow cover, is derived from the existing integer flow cover inequalities by a lifting procedure. The lifting coefficients are sequence independent when the batch sizes (V) and the production capacities (P) are constant and when V divides P. When the restriction of the divisibility is removed, the lifting coefficients are shown to be sequence independent. We identify some cases where the mixed flow cover inequalities are facet defining. We propose a cutting plane algorithm for different classes of valid inequalities introduced in the paper. The exact separation algorithm proposed for the constant capacitated case runs in polynomial time. Computational results show the efficiency of the new class mixed flow cover compared to the existing methods.     

A polyhedral approach to the alldifferent system

This paper examines the facial structure of the convex hull of integer vectors satisfying a system of alldifferent predicates, also called an alldifferent system. The underlying analysis is based on a property, called inclusion, pertinent to such a system. For the alldifferent systems for which this property holds, we present two families of facet-defining inequalities, establish that they completely describe the convex hull and show that they can be separated in polynomial time. Consequently, the inclusion property characterises a group of alldifferent systems for which the linear optimization problem (i.e. the problem of optimizing a linear function over that system) can be solved in polynomial time. Furthermore, we establish that, for systems with three predicates, the inclusion property is also a necessary condition for the convex hull to be described by those two families of inequalities. For the alldifferent systems that do not possess that property, we establish another family of facet-defining inequalities and an accompanied polynomial-time separation algorithm. All the separation algorithms are incorporated within a cutting-plane scheme and computational experience on a set of randomly generated instances is reported. In concluding, we show that the pertinence of the inclusion property can be decided in polynomial time.     

n-step mingling inequalities: new facets for the mixed-integer knapsack set

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.     

Higher-order cover cuts from zero–one knapsack constraints augmented by two-sided bounding inequalities

Extending our work on second-order cover cuts [F. Glover, H.D. Sherali, Second-order cover cuts, Mathematical Programming (ISSN: 0025-5610 1436-4646) (2007), doi:10.1007/s10107-007-0098-4. (Online)], we introduce a new class of higher-order cover cuts that are derived from the implications of a knapsack constraint in concert with supplementary two-sided inequalities that bound the sums of sets of variables. The new cuts can be appreciably stronger than the second-order cuts, which in turn dominate the classical knapsack cover inequalities. The process of generating these cuts makes it possible to sequentially utilize the second-order cuts by embedding them in systems that define the inequalities from which the higher-order cover cuts are derived. We characterize properties of these cuts, design specialized procedures to generate them, and establish associated dominance relationships. These results are used to devise an algorithm that generates all non-dominated higher-order cover cuts, and, in particular, to formulate and solve suitable separation problems for deriving a higher-order cut that deletes a given fractional solution to an underlying continuous relaxation. We also discuss a lifting procedure for further tightening any generated cut, and establish its polynomial-time operation for unit-coefficient cuts. A numerical example is presented that illustrates these procedures and the relative strength of the generated non-redundant, non-dominated higher-order cuts, all of which turn out to be facet-defining for this example. Some preliminary computational results are also presented to demonstrate the efficacy of these cuts in comparison with lifted minimal cover inequalities for the underlying knapsack polytope.     

Mingling: mixed-integer rounding with bounds

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.     

Generalized cover facet inequalities for the generalized assignment problem

The generalized assignment problem is that of finding an optimal assignment of agents to tasks, where each agent may be assigned multiple tasks and each task is performed exactly once. This is an NP-complete problem. Algorithms that employ information about the polyhedral structure of the associated polytope are typically more effective for large instances than those that ignore the structure. A class of generalized cover facet-defining inequalities for the generalized assignment problem is derived. These inequalities are based upon multiple knapsack constraints and are derived from generalized cover inequalities.