on the exact separation of mixed integer knapsack cuts during the last decades much research has been conducted on deriving classes of valid inequalities for mixed integer knapsack sets which we call knapsack cuts bixby et al the sharpest cut the impact of manfred padberg and his work mps siam series on optimization pp 309 326 2004 empirically observe that within the context of branch and cut algorithms to solve mixed integer programming problems the most important inequalities are knapsack cuts derived by the mixed integer rounding mir procedure in this work we analyze this empirical observation by developing an algorithm to separate over the convex hull of a mixed integer knapsack set the main feature of this algorithm is a specialized subroutine for optimizing over a mixed integer knapsack set which exploits dominance relationships the exact separation of knapsack cuts allows us to establish natural benchmarks by which to evaluate specific classes of them using these benchmarks on miplib 3 0 and miplib 2003 instances we analyze the performance of mir inequalities our computations which are performed in exact arithmetic are surprising in the vast majority of the instances in which knapsack cuts yield bound improvements mir cuts alone achieve over 87 of the observed gain

aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

on the separation of disjunctive cuts disjunctive cuts for mixed integer linear programs mips were introduced by egon balas in the late 1970s and have been successfully exploited in practice since the late 1990s in this paper we investigate the main ingredients of a disjunctive cut separation procedure and analyze their impact on the quality of the root node bound for a set of instances taken from miplib library we compare alternative normalization conditions and try to better understand their role in particular we point out that constraints that become redundant because of the disjunction used can produce over weak cuts and analyze this property with respect to the normalization used finally we introduce a new normalization condition and analyze its theoretical properties and computational behavior along the way we make use of a number of small numerical examples to illustrate some basic and often misinterpreted disjunctive programming features

pivot and reduce cuts an approach for improving gomory mixed integer cuts abstract gomory mixed integer cuts are of crucial importance in solving large scale mixed integer linear programs recently there has been considerable research particularly on the strengthening of these cuts we present a new heuristic algorithm which aims at improving gomory mixed integer cuts our approach is related to the reduce and split cuts these cuts are based on an algorithm which tries to reduce the coefficients of the continuous variables by forming integer linear combinations of simplex tableau rows our algorithm is designed to achieve the same result by performing pivots on the simplex tableau we give a detailed description of the algorithm and its implementation finally we report on computational results with our approach and analyze its performance the results indicate that our algorithm can enhance the performance of the gomory mixed integer cuts

cut generation through binarization for a mixed integer linear program where all integer variables are bounded we study a reformulation introduced by roy that maps general integer variables to a collection of binary variables we study theoretical properties and empirical strength of rank 2 simple split cuts of the reformulation we show that for a pure integer problem with two integer variables these cuts are sufficient to obtain the integer hull of the problem but that this does not generalize to problems in higher dimensions we also give an algorithm to compute an approximation of the rank 2 simple split cut closure we report empirical results on 22 benchmark instances showing that the bounds obtained compare favorably with those obtained with other approximate methods to compute the split closure or lattice free cut closure it gives a better bound than the split closure on 6 instances while it is weaker on 9 instances for an average gap closed 3 8 smaller than the one for the split closure

binary extended formulations of polyhedral mixed integer sets we analyze different ways of constructing binary extended formulations of polyhedral mixed integer sets with bounded integer variables and compare their relative strength with respect to split cuts we show that among all binary extended formulations where each bounded integer variable is represented by a distinct collection of binary variables what we call unimodular extended formulations are the strongest we also compare the strength of some binary extended formulations from the literature finally we study the behavior of branch and bound on such extended formulations and show that branching on the new binary variables leads to significantly smaller enumeration trees in some cases

mixed integer sets from two rows of two adjacent simplex bases in andersen et al lecture notes in computer science vol 4513 springer berlin pp 1 15 2007 we studied a mixed integer set arising from two rows of a simplex tableau we showed that facets of such a set can be obtained from lattice point free triangles and quadrilaterals associated with either three or four variables in this paper we generalize our findings and show that when upper bounds on the non basic variables are also considered further classes of facets arise that cannot be obtained from triangles and quadrilaterals specifically when exactly one upper bound on a non basic variable is introduced stronger inequalities that can be derived from pentagons involving up to six variables also appear

valid inequalities for mixed integer linear programs this tutorial presents a theory of valid inequalities for mixed integer linear sets it introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift and project cuts gomory mixed integer cuts mixed integer rounding cuts split cuts and intersection cuts and it reveals the relationships between these families the tutorial also discusses computational aspects of generating the cuts and their strength

controllability of kinematic control systems on stratified configuration spaces this paper considers nonlinear kinematic controllability of a class of systems called stratified roughly speaking such stratified systems have a configuration space which can be decomposed into sub manifolds upon which the system has different sets of equations of motion for such systems considering the controllability is difficult because of the discontinuous form of the equations of motion the main result in this paper is a controllability test analogous to chow s theorem is based upon a construction involving distributions and the extension thereof to robotic gaits

split closure and intersection cuts in the seventies balas introduced intersection cuts for a mixed integer linear program milp and showed that these cuts can be obtained by a closed form formula from a basis of the standard linear programming relaxation in the early nineties cook kannan and schrijver introduced the split closure of a milp and showed that the split closure is a polyhedron in this paper we show that the split closure can be obtained using only intersection cuts we give two different proofs of this result one geometric and one algebraic the result is then used to provide a new proof of the fact that the split closure of a milp is a polyhedron finally we extend the result to more general disjunctions

bilevel programming and the separation problem in recent years branch and cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs milps methods for automatically generating inequalities valid for the convex hull of solutions to such milps are a critical element of branch and cut this paper examines the nature of the so called separation problem which is that of generating a valid inequality violated by a given real vector usually arising as the solution to a relaxation of the original problem we show that the problem of generating a maximally violated valid inequality often has a natural interpretation as a bilevel program in some cases this bilevel program can be easily reformulated as a simple single level mathematical program yielding a standard mathematical programming formulation for the separation problem in other cases no such polynomial size single level reformulation exists unless the polynomial hierarchy collapses to its first level an event considered extremely unlikely in computational complexity theory we illustrate our insights by considering the separation problem for two well known classes of valid inequalities

on mixing inequalities rank closure and cutting plane proofs we study the mixing inequalities that were introduced by gunluk and pochet math program 90 2001 pp 429 457 we show that a mixing inequality which mixes n mir inequalities has mir rank at most n if it is a type i mixing inequality and at most n 1 if it is a type ii mixing inequality we also show that these bounds are tight for n 2 given a mixed integer set p i p cap z i where p is a polyhedron and z i x in mathbb r n x i in mathbb z forall i in i we define mixing inequalities for p i we show that the elementary mixing closure of p with respect to i can be described using a bounded number of mixing inequalities each of which has a bounded number of terms this implies that the elementary mixing closure of p is a polyhedron finally we show that any mixing inequality can be derived via a polynomial length mir cutting plane proof combined with results of dash on the complexity of cutting plane proofs using split cuts ibm research report rc 24082 oct 2006 and pudlak j symbolic logic 62 1997 pp 981 998 this implies that there are valid inequalities for a certain mixed integer set that cannot be obtained via a polynomial size mixing cutting plane proof

on t branch split cuts for mixed integer programs in this paper we study the t branch split cuts introduced by li and richard discret optim 5 724 734 2008 they presented a family of mixed integer programs with n integer variables and a single continuous variable and conjectured that the convex hull of integer solutions for any n has unbounded rank with respect to n 1 branch split cuts it was shown earlier by cook et al math program 47 155 174 1990 that this conjecture is true when n 2 and li and richard proved the conjecture when n 3 in this paper we show that this conjecture is also true for all n 3

generalized chvatal gomory closures for integer programs with bounds on variables integer programming problems that arise in practice often involve decision variables with one or two sided bounds in this paper we consider a generalization of chvatal gomory inequalities obtained by strengthening chvatal gomory inequalities using the bounds on the variables we prove that the closure of a rational polyhedron obtained after applying the generalized chvatal gomory inequalities is also a rational polyhedron this generalizes a result of dunkel and schulz on 0 1 problems to the case when some of the variables have upper or lower bounds or both while the rest of them are unbounded

intersection cuts with infinite split rank we consider mixed integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables when this model only has two integer variables dey and louveaux characterized the intersection cuts that have infinite split rank we show that for any number of integer variables the split rank of an intersection cut generated from a rational lattice free polytope l is finite if and only if the integer points on the boundary of l satisfy a certain 2 hyperplane property the dey louveaux characterization is a consequence of this more general result

a recursive procedure to generate all cuts for 0 1 mixed integer programs we study several ways of obtaining valid inequalities for mixed integer programs we show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to gomory s mixed integer cuts we also show how all valid inequalities for mixed 0 1 programs can be generated recursively from a simple subclass of the disjunctive inequalities

mixed integer quadratic programming is in np mixed integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral in this paper we prove that the decision version of mixed integer quadratic programming is in np thereby showing that it is np complete this is established by showing that if the decision version of mixed integer quadratic programming is feasible then there exists a solution of polynomial size this result generalizes and unifies classical results that quadratic programming is in np vavasis in inf process lett 36 2 73 77 17 and integer linear programming is in np borosh and treybig in proc am math soc 55 299 304 1 von zur gathen and sieveking in proc am math soc 72 155 158 18 kannan and monma in lecture notes in economics and mathematical systems vol 157 pp 161 172 springer 9 papadimitriou in j assoc comput mach 28 765 768 15

cutting planes for multistage stochastic integer programs this paper addresses the problem of finding cutting planes for multistage stochastic integer programs we give a general method for generating cutting planes for multistage stochastic integer programs based on combining inequalities that are valid for the individual scenarios we apply the method to generate cuts for a stochastic version of a dynamic knapsack problem and for stochastic lot sizing problems we give computational results which show that these new inequalities are very effective in a branch and cut algorithm

a 3 slope theorem for the infinite relaxation in the plane in this paper we consider the infinite relaxation of the corner polyhedron with 2 rows for the 1 row case gomory and johnson proved in their seminal paper a sufficient condition for a minimal function to be extreme the celebrated 2 slope theorem despite increased interest in understanding the multiple row setting no generalization of this theorem was known for this case we present an extension of the 2 slope theorem for the case of 2 rows by showing that minimal 3 slope functions satisfying an additional regularity condition are facets and hence extreme moreover we show that this regularity condition is necessary unveiling a structure which is only present in the multi row setting

new modeling approaches for the design of local access transport area networks abstract this paper is concerned with the design of a local access and transport area lata network that arises in the context of providing a special access service by a local exchange facility the problem includes locating suitable hub facilities having adequate multiplexing capacity and routing the demand traffic from various clients end offices through their hubs or alternatively through a special point of presence node in order to minimize total costs these costs involve various traffic distribution and equipment component costs where the latter typically exhibit nonlinear economies of scale we develop a model for this problem and apply the reformulation linearization technique rlt to construct various enhanced tightened versions of the proposed model we also design a hybrid heuristic procedure that combines a limited run of an exact method with a lagrangian dual relaxation procedure extensive computational results are provided to study the relative tightness resulting from the various formulations and their effect on providing good quality feasible solutions

on the relative strength of families of intersection cuts arising from pairs of tableau constraints in mixed integer programs we compare the relative strength of valid inequalities for the integer hull of the feasible region of mixed integer linear programs with two equality constraints two unrestricted integer variables and any number of nonnegative continuous variables in particular we prove that the closure of type 2 triangle resp type 3 triangle quadrilateral inequalities are all within a factor of 1 5 1 5 of the integer hull and provide examples showing that the approximation factor is not less than 1 125 1 125 there is no fixed approximation ratio for split or type 1 triangle inequalities however

valid inequalities and facets of the capacitated plant location problem recently several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods and polyhedral characterizations of integer programming problems in this paper we investigate the polyhedral structure of the capacitated plant location problem our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure

design and verify a new scheme for generating cutting planes a cutting plane procedure for integer programming ip problems usually involves invoking a black box procedure such as the gomory chvatal procedure to compute a cutting plane in this paper we describe an alternative paradigm of using the same cutting plane black box this involves two steps in the first step we design an inequality cx le d where c and d are integral independent of the cutting plane black box in the second step we verify that the designed inequality is a valid inequality by verifying that the set p cap x in mathbb r n mid cx ge d 1 cap mathbb z n is empty using cutting planes from the black box here p is the feasible region of the linear programming relaxation of the ip we refer to the closure of all cutting planes that can be verified to be valid using a specific cutting plane black box as the verification closure of the considered cutting plane black box this paper undertakes a systematic study of properties of verification closures of various cutting plane black box procedures

finite disjunctive programming characterizations for general mixed integer linear programs in this paper we give a finite disjunctive programming procedure to obtain the convex hull of general mixed integer linear programs milp with bounded integer variables we propose a finitely convergent convex hull tree algorithm that constructs a linear program that has the same optimal solution as the associated milp in addition we combine the standard notion of sequential cutting planes with ideas underlying the convex hull tree algorithm to help guide the choice of disjunctions to use within a cutting plane method this algorithm which we refer to as the cutting plane tree algorithm is shown to converge to an integral optimal solution in finitely many iterations finally we illustrate the proposed algorithm on three well known examples in the literature that require an infinite number of elementary or split disjunctions in a rudimentary cutting plane algorithm

reverse split rank the reverse split rank of an integral polyhedron p p is defined as the supremum of the split ranks of all rational polyhedra whose integer hull is p p already in mathbb r 3 r3 there exist polyhedra with infinite reverse split rank we give a geometric characterization of the integral polyhedra in mathbb r n rn with infinite reverse split rank

a convex analysis perspective on disjunctive cuts we treat with tools from convex analysis the general problem of cutting planes separating a point from a closed convex set p crucial for this is the computation of extreme points in the so called reverse polar set introduced by e balas in 1979 in the polyhedral case this enables the computation of cuts that define facets of p we exhibit three equivalent optimization problems to compute such extreme points one of them corresponds to selecting a specific normalization to generate cuts we apply the above development to the case where p is the closed convex hull of a union and more particularly a union of polyhedra case of disjunctive cuts we conclude with some considerations on the design of efficient cut generators the paper also contains an appendix reviewing some fundamental concepts of convex analysis

reduce and split cuts improving the performance of mixed integer gomory cuts mixed integer gomory cuts have become an integral part of state of the art software for solving mixed integer linear programming problems therefore improvements in the performance of these cutting planes can be of great practical value in this paper we present a simple and fast heuristic for improving the coefficients on the continuous variables in the mixed integer gomory cuts this is motivated by the fact that in a mixed integer gomory cut the coefficient of an integer variable lies between 0 and 1 whereas for a continuous variable there is no upper bound the heuristic tries to reduce the coefficients of the continuous variables we call the resulting cuts reduce and split cuts we found that on several test problems reduce and split cuts can substantially enhance the performance of a branch and bound algorithm

notions of maximality for integral lattice free polyhedra the case of dimension three lattice free sets and their applications for cutting plane methods in mixed integer optimization have been studied in recent literature the family of all integral lattice free polyhedra that are not properly contained in another integral lattice free polyhedron has been of particular interest we call these polyhedra d maximal for fixed d the family of d maximal integral lattice free polyhedra is finite up to unimodular equivalence in view of possible applications in cutting plane theory one would like to have a classification of this family this is a challenging task already for small dimensions in contrast the subfamily of all integral lattice free polyhedra that are not properly contained in any other lattice free set which we call d maximal lattice free polyhedra allow a rather simple geometric characterization hence the question was raised for which dimensions the notions of d maximality and d maximality are equivalent this was known to be the case for dimensions one and two on the

the mixing mir set with divisible capacities we study the set s x y in re times z n x b j y j geq b j j 1 ldots n where b j b j in re 0 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 gunluk 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

convex hull of two quadratic or a conic quadratic and a quadratic inequality in this paper we consider an aggregation technique introduced by yildiran j math control inf 26 417 450 2009 to study the convex hull of regions defined by two quadratic inequalities or by a conic quadratic and a quadratic inequality yildiran 2009 shows how to characterize the convex hull of open sets defined by two strict quadratic inequalities using linear matrix inequalities we show how this aggregation technique can be easily extended to yield valid conic quadratic inequalities for the convex hull of open sets defined by two strict quadratic inequalities or by a strict conic quadratic and a strict quadratic inequality we also show that for sets defined by a strict conic quadratic and a strict quadratic inequality under one additional containment assumption these valid inequalities characterize the convex hull exactly we also show that under certain topological assumptions the results from the open setting can be extended to characterize the closed convex hull of sets defined with non strict conic and quadratic inequalities

a connection between cutting plane theory and the geometry of numbers in this paper we relate several questions about cutting planes to a fundamental problem in the geometry of numbers namely the closest vector problem using this connection we show that the dominance membership and validity problems are np complete for chvatal and split cuts

two dimensional lattice free cuts and asymmetric disjunctions for mixed integer polyhedra in this paper we study the relationship between 2d lattice free cuts the family of cuts obtained by taking two row relaxations of a mixed integer program mip and applying intersection cuts based on maximal lattice free sets in r 2 and various types of disjunctions recently li and richard 2007 studied disjunctive cuts obtained from t branch split disjunctions of mixed integer sets these cu ts generalize split cuts balas 2009 initiated the study of cuts for the two row continuous group relaxation obtained from 2 branch split disjunctions we study these cuts and call them cross cuts for the two row continuous group relaxation and for general mips we also consider cuts obtained from asymmetric 2 branch disjunctions which we call crooked cross cuts for the two row continuous group relaxation we show that unimodular cross cuts the coefficients of the two split inequalities fo rm a unimodular matrix are equivalent to the cuts obtained from maximal lattice free sets other than type 3 triangles we also prove that all 2d lattice free cuts and their s free extensions ar e crooked cross cuts for general mixed integer sets we show that crooked cross cuts can be generated from a structured three row relaxation finally we show that for the corner relaxation of an mip every crooked cross cut is a 2d lattice free cut

a new lift and project operator abstract in this paper we analyze the strength of split cuts in a lift and project framework we first observe that the lovasz schrijver and sherali adams lift and project operator hierarchies can be viewed as applying specific 0 1 split cuts to an appropriate extended formulation and demonstrate how to strengthen these hierarchies using additional split cuts more precisely we define a new operator that adds all 0 1 split cuts to the extended formulation for 0 1 mixed integer sets with k binary variables this new operator is guaranteed to obtain the integer hull in k 2 steps compared to k steps for the lovasz schrijver or the sherali adams operator we also present computational results on the stable set problem with our new operator

sequence independent lifting for mixed integer programming we show that superadditive lifting functions lead to sequence independent lifting of inequalities for general mixed integer programming as an application we note that mixed integer rounding mir may be viewed as sequence independent lifting consequently we obtain facet conditions for mir inequalities for mixed integer knapsacks

lattice closures of polyhedra given p subset mathbb r n a mixed integer set p i p cap mathbb z t times mathbb r n t and a k tuple of n dimensional integral vectors pi 1 ldots pi k where the last n t entries of each vector is zero we consider the relaxation of p i obtained by taking the convex hull of points x in p for which pi 1 tx ldots pi t kx are integral we then define the k dimensional lattice closure of p i to be the intersection of all such relaxations obtained from k tuples of n dimensional vectors when p is a rational polyhedron we show that given any collection of such k tuples there is a finite subcollection that gives the same closure more generally we show that any k tuple is dominated by another k tuple coming from the finite subcollection the k dimensional lattice closure contains the convex hull of p i and is equal to the split closure when k 1 therefore a result of cook et al math program 47 155 174 1990 implies that when p is a rational polyhedron the k dimensional lattice closure is a polyhedron for k 1 and our finiteness result extends this to all k ge 2 we also construct a polyhedral mixed integer set with n integer variables and one continuous variable such that for any k n finitely many iterations of the k dimensional lattice closure do not give the convex hull of the set our result implies that t branch split cuts cannot give the convex hull of the set nor can valid inequalities from unbounded full dimensional convex lattice free sets

intersection cuts for nonlinear integer programming convexification techniques for structured sets we study the generalization of split k branch split and intersection cuts from mixed integer linear programming to the realm of mixed integer nonlinear programming constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure we introduce two techniques to give precise characterizations of such convex hulls and use them to construct split k branch split and intersection cuts for several classes of non polyhedral sets in particular we give simple formulas for split cuts for essentially all convex sets described by a single conic quadratic inequality we also give simple formulas for k branch split cuts and some general intersection cuts for a wide variety of convex quadratic sets

lattice free sets multi branch split disjunctions and mixed integer programming in this paper we study the relationship between valid inequalities for mixed integer sets lattice free sets associated with these inequalities and the multi branch split cuts introduced by li and richard discret optim 5 724 734 2008 by analyzing n dimensional lattice free sets we prove that for every integer n there exists a positive integer t such that every facet defining inequality of the convex hull of a mixed integer polyhedral set with n integer variables is a t branch split cut we use this result to give a finite cutting plane algorithm to solve mixed integer programs we also show that the minimum value t for which all facets of polyhedral mixed integer sets with n integer variables can be generated as t branch split cuts grows exponentially with n in particular when n 3 we observe that not all facet defining inequalities are 6 branch split cuts

the cutting plane method is polynomial for perfect matchings the cutting plane approach to finding minimum cost perfect matchings has been discussed by several authors over past decades its convergence has been an open question we develop a cutting plane algorithm that converges in polynomial time using only edmonds blossom inequalities and which maintains half integral intermediate lp solutions supported by a disjoint union of odd cycles and edges our main insight is a method to retain only a subset of the previously added cutting planes based on their dual values this allows us to quickly find violated blossom inequalities and argue convergence by tracking the number of odd cycles in the support of intermediate solutions

the continuous mixing polyhedron we analyze the polyhedral structure of the setspcmix s r z r xr nxz n s r j z j f j j 1 n andp cmix pcmixn s 0 the setp cmix is a natural generalization of the mixing set studied by pochet and wolsey 15 16 and ga nla k and pochet 8 and recently has been introduced by miller and wolsey 12 we introduce a new class of valid inequalities that has proven to be sufficient for describing conv pcmix we give an extended formulation of sizeo n xo n2 variables and constraints and indicate how to separate over conv pcmix ino n3 time finally we show how the mixed integer rounding mir inequalities of nemhauser and wolsey 14 and the mixing inequalities of ga nla k and pochet 8 constitute special cases of the cycle inequalities

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

on the uncapacitated plant location problem i valid inequalities and facets the uncapacitated plant location problem is considered as a node packing problem for this problem several valid inequalities and facets are discussed necessary and sufficient conditions for trivial facets along with necessary conditions for nontrivial facets are derived in addition all of the facets for the case of three plants and three or more destinations are identified

a precise correspondence between lift and project cuts simple disjunctive cuts and mixed integer gomory cuts for 0 1 programming we establish a precise correspondence between lift and project cuts for mixed 0 1 programs simple disjunctive cuts intersection cuts and mixed integer gomory cuts the correspondence maps members of one family onto members of the others it also maps bases of the higher dimensional cut generating linear program cglp into bases of the linear programming relaxation it provides new bounds on the number of facets of the elementary closure and on the rank of the standard linear programming relaxation of the mixed 0 1 polyhedron with respect to the above families of cutting planes

the rank of mixed integer polyhedra we define a purely geometrical notion of the rank of mixed integer rational polyhedra that differs substantially from the existing notions found in the literature

on the enumerative nature of gomory s dual cutting plane method for 30 years after their invention half a century ago cutting planes for integer programs have been an object of theoretical investigations that had no apparent practical use when they finally proved their practical usefulness in the late eighties that happened in the framework of branch and bound procedures as an auxiliary tool meant to reduce the number of enumerated nodes to this day pure cutting plane methods alone have poor convergence properties and are typically not used in practice our reason for studying them is our belief that these negative properties can be understood and thus remedied only based on a thorough investigation of such procedures in their pure form in this paper the second in a sequence we address some important issues arising when designing a computationally sound pure cutting plane method we analyze the dual cutting plane procedure proposed by gomory in 1958 which is the first and most famous convergent cutting plane method for integer linear programming we focus on the enumerative nature of this method as evidenced by the relative computational success of its lexicographic version as documented in our previous paper on the subject and we propose new versions of gomory s cutting plane procedure with an improved performance in particular the new versions are based on enumerative schemes that treat the objective function implicitly and redefine the lexicographic order on the fly to mimic a sound branching strategy preliminary computational results are reported

on the complexity of selecting disjunctions in integer programming the imposition of general disjunctions of the form pi x leq pi 0 vee pi x geq pi 0 1 where pi pi 0 are integer valued is a fundamental operation in both the branch and bound and cutting plane algorithms for solving mixed integer linear programs such disjunctions can be used for branching at each iteration of the branch and bound algorithm or to generate split inequalities for the cutting plane algorithm we first consider the problem of selecting a general disjunction and show that the problem of selecting an optimal such disjunction according to specific criteria described herein is mathcal np hard we further show that the problem remains mathcal np hard even for binary programs or when considering certain restricted classes of disjunctions we observe that the problem of deciding whether a given inequality is a split inequality can be reduced to one of the above problems which leads to a proof that the problem is mathcal np complete

facets of the bipartite subgraph polytope the bipartite subgraph polytope pbg of a graph g v e is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of g we show that all complete subgraphs of g of odd order and all so called odd bicycle wheels contained in g induce facets of pbg moreover we describe several methods with which new facet defining inequalities of pbg can be constructed from known ones examples of these methods are contraction of node sets in odd complete subgraphs odd subdivision of edges certain splittings of nodes and subdivision of all edges of a cut using these methods we can construct facet defining inequalities of pbg having coefficients of order v 2

projected chvatal gomory cuts for mixed integer linear programs recent experiments by fischetti and lodi show that the first chvatal closure of a pure integer linear program ilp often gives a surprisingly tight approximation of the integer hull they optimize over the first chvatal closure by modeling the chvatal gomory cg separation problem as a mixed integer linear program milp which is then solved by a general purpose milp solver unfortunately this approach does not extend immediately to the gomory mixed integer gmi closure of an milp since the gmi separation problem involves the solution of a nonlinear mixed integer program or a parametric milp in this paper we introduce a projected version of the cg cuts and study their practical effectiveness for milp problems the idea is to project first the linear programming relaxation of the milp at hand onto the space of the integer variables and then to derive chvatal gomory cuts for the projected polyhedron though theoretically dominated by gmi cuts projected cg cuts have the advantage of producing a separation model very similar to the one introduced by fischetti and lodi which can typically be solved in a reasonable amount of computing time

on the facets of mixed integer programs with two integer variables and two constraints in this paper we consider an infinite relaxation of the mixed integer linear program with two integer variables nonnegative continuous variables and two equality constraints and we give a complete characterization of its facets we also derive an analogous characterization of the facets of the underlying finite integer program

branching on general disjunctions this paper considers a modification of the branch and cut algorithm for mixed integer linear programming where branching is performed on general disjunctions rather than on variables we select promising branching disjunctions based on a heuristic measure of disjunction quality this measure exploits the relation between branching disjunctions and intersection cuts in this work we focus on disjunctions defining the mixed integer gomory cuts at an optimal basis of the linear programming relaxation the procedure is tested on instances from the literature experiments show that for a majority of the instances the enumeration tree obtained by branching on these general disjunctions has a smaller size than the tree obtained by branching on variables even when variable branching is performed using full strong branching

minimal valid inequalities for integer constraints in this paper we consider a semi infinite relaxation of mixed integer linear programs we show that minimal valid inequalities for this relaxation correspond to maximal lattice free convex sets and that they arise from nonnegative piecewise linear positively homogeneous convex functions

split rank of triangle and quadrilateral inequalities a simple relaxation consisting of two rows of a simplex tableau is a mixed integer set with two equations two free integer variables and nonnegative continuous variables recently andersen et al and cornuejols and margot showed that the facet defining inequalities of this set are either split cuts or intersection cuts obtained from lattice free triangles and quadrilaterals from an example given by cook et al it is known that one particular class of facet defining triangle inequality does not have finite split rank in this paper we show that all other facet defining triangle and quadrilateral inequalities have finite split rank

on cutting plane proofs in combinatorial optimization abstract gomory s cutting plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron the number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron we prove that polyhedra featured in popular formulations of the stable set problem the set covering problem the set partitioning problem the knapsack problem the bipartite subgraph problem the maximum cut problem the acyclic subdigraph problem the asymmetric traveling salesman problem and the traveling salesman problem have arbitrarily high rank in particular we prove conjectures of barahona grotschel and mahjoub chvatal grotschel and pulleyblank and junger

on the rank of disjunctive cuts let l be a family of lattice free polyhedra in rm containing the splits given a polyhedron p in rm n we characterize when a valid inequality for p zm rn can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in l we also characterize the lattice free polyhedra m such that all the disjunctive cuts corresponding to m can be obtained with a finite number of disjunctive cuts corresponding to the polyhedra in l for every polyhedron p our results imply interesting consequences related to split rank and to integral lattice free polyhedra that extend recent research findings

using cutting planes in an interactive reference point approach for multiobjective integer linear programming problems we propose an interactive approach for multiple objective integer linear programming moilp problems that combines the use of the tchebycheff metric with cutting plane techniques at each interaction the method computes the nondominated solution for the moilp problem that is closest to a reference point according to the tchebycheff metric the information provided by the decision maker in each dialogue phase is used to adjust the next reference point through a sensitivity analysis stage cutting plane techniques enable the method to take advantage of computations performed at previous iterations to solve the next scalarizing integer program we address both theoretical issues and the computational implementation

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

mir closures of polyhedral sets we study the mixed integer rounding mir closures of polyhedral sets the mir closure of a polyhedral set is equal to its split closure and the associated separation problem is np hard we describe a mixed integer programming mip model with linear constraints and a non linear objective for separating an arbitrary point from the mir closure of a given mixed integer set we linearize the objective using additional variables to produce a linear mip model that solves the separation problem exactly using a subset of these additional variables yields an mip model which solves the separation problem approximately with an accuracy that depends on the number of additional variables used our analysis yields an alternative proof of the result of cook et al 1990 that the split closure of a polyhedral set is again a polyhedron we also discuss a heuristic to obtain mir cuts based on our approximate separation model and present some computational results

unbounded convex sets for non convex mixed integer quadratic programming this paper introduces a fundamental family of unbounded convex sets that arises in the context of non convex mixed integer quadratic programming it is shown that any mixed integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family some fundamental properties of the convex sets are derived along with connections to some other well studied convex sets several classes of valid and facet inducing inequalities are also derived

clique tree inequalities and the symmetric travelling salesman problem the linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful cf crowder and padberg crowder h p m w padberg 1980 solving large scale symmetric travelling salesman problems to optimality management sci 26 495 509 grotschel grotschel m 1980a on the symmetric travelling salesman problem solution of a 120 city problem math programming stud 12 61 77 padberg and hong padberg m w s hong 1980 on the symmetric travelling salesman problem a computational study math programming stud 12 78 107 one of the reasons for this success is certainly the fact that instead of ordinary cutting planes gomory cuts etc problem specific cutting planes could be used which define facets of the underlying integer programming polytopes in this paper we shall define a new class of inequalities clique tree inequalities valid for the travelling salesman polytope which properly contains many of the known classes of inequalities like subtour elimination constraints 2 matching constraints comb inequalities and we show that all these new inequalities induce facets of the travelling salesman polytope since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem

generalized intersection cuts and a new cut generating paradigm intersection cuts are generated from a polyhedral cone and a convex set s whose interior contains no feasible integer point we generalize these cuts by replacing the cone with a more general polyhedron c the resulting generalized intersection cuts dominate the original ones this leads to a new cutting plane paradigm under which one generates and stores the intersection points of the extreme rays of c with the boundary of s rather than the cuts themselves these intersection points can then be used to generate in a non recursive fashion cuts that would require several recursive applications of some standard cut generating routine a procedure is also given for strengthening the coefficients of the integer constrained variables of a generalized intersection cut the new cutting plane paradigm yields a new characterization of the closure of intersection cuts and their strengthened variants this characterization is minimal in the sense that every one of the inequalities it uses defines a facet of the closure

the involution width of finite simple groups abstract for a finite group generated by involutions the involution width is defined to be the minimal k n such that any group element can be written as a product of at most k involutions we show that the involution width of every non abelian finite simple group is at most 4 this result is sharp as there are families with involution width precisely 4

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 atamturk 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 gunluk math program 105 29 53 2006 under some conditions

two row mixed integer cuts via lifting recently andersen et al 1 borozan and cornuejols 6 and cornuejols and margot 9 have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non negative continuous variables these inequalities are either split cuts or intersection cuts derived using maximal lattice free convex sets in order to use these inequalities to obtain cuts from two rows of a general simplex tableau one approach is to extend the system to include all possible non negative integer variables giving the two row mixed integer infinite group problem and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities in this paper we study the characteristics of these lifting functions we show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice free triangle with multiple integer points in the relative interior of one of its sides or a maximal lattice free triangle with integral vertices and one integer point in the relative interior of each side in the other cases maximal lattice free triangles with one integer point in the relative interior of each side and non integral vertices and maximal lattice free quadrilaterals non unique lifting functions may yield distinct extreme inequalities for the latter family of triangles we present sufficient conditions to yield an extreme inequality for the two row mixed integer infinite group problem

a note on the split rank of intersection cuts in this note we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts as a first step of this argument a polyhedral subset of the lattice free convex set that is used to generate the intersection cut is constructed we call this subset the restricted lattice free set it is then shown that lceil log 2 l rceil is a lower bound on the split rank of the intersection cut where l is the number of integer points lying on the boundary of the restricted lattice free set satisfying the condition that no two points lie on the same facet of the restricted lattice free set the use of this result is illustrated by obtaining a lower bound of lceil log 2 n 1 rceil on the split rank of n row mixing inequalities

on convergence in mixed integer programming let p subseteq mathbb r m n be a rational polyhedron and let pi be the convex hull of p cap mathbb z m times mathbb r n we define the integral lattice free closure of p as the set obtained from p by adding all inequalities obtained from disjunctions associated with integral lattice free polyhedra in mathbb r m we show that the integral lattice free closure of p is again a polyhedron and that repeatedly taking the integral lattice free closure of p gives pi after a finite number of iterations such results can be seen as a mixed integer analogue of theorems by chvatal and schrijver for the pure integer case one ingredient of our proof is an extension of a result by owen and mehrotra in fact we prove that for each rational polyhedron p the split closures of p yield in the limit the set pi

on the polyhedrality of cross and quadrilateral closures split cuts form a well known class of valid inequalities for mixed integer programming problems cook et al math program 47 155 174 1990 showed that the split closure of a rational polyhedron p is again a polyhedron in this paper we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra we then use this result to prove that cross cuts yield closures that are rational polyhedra cross cuts are a generalization of split cuts introduced by dash et al math program 135 221 254 2012 finally we show that the quadrilateral closure of the two row continuous group relaxation is a polyhedron answering an open question in basu et al math program 126 281 314 2011

cutting planes from extended lp formulations given a mixed integer set defined by linear inequalities and integrality requirements on some of the variables we consider extended formulations of its continuous lp relaxation and study the effect of adding cutting planes in the extended space in terms of optimization extended lp formulations do not lead to better bounds as their projection onto the original space is precisely the original lp relaxation however adding cutting planes in the extended space can lead to stronger bounds in this paper we show that for every 0 1 mixed integer set with n integer and k continuous variables there is an extended lp formulation with 2n k 1 2n k 1 variables whose elementary split closure is integral the proof is constructive but it requires an inner description of the lp relaxation we then extend this idea to general mixed integer sets and construct the best extended lp formulation for such sets with respect to lattice free cuts we also present computational results on the two row continuous group relaxation showing the strength of cutting planes derived from extended lp formulations

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection

optimizing over the split closure the polyhedron defined by all the split cuts obtainable directly i e without iterated cut generation from the lp relaxation p of a mixed integer program mip is termed the elementary or rank 1 split closure of p this paper deals with the problem of optimizing over the elementary split closure this is accomplished by repeatedly solving the following separation problem given a fractional point say x find a rank 1 split cut violated by x or show that none exists following caprara and letchford 17 we formulate this separation problem as a nonlinear mixed integer program that can be treated as a parametric mixed integer linear program pmilp with a single parameter in the objective function and the right hand side we develop an algorithmic framework to deal with the resulting pmilp by creating and maintaining a dynamically updated grid of parameter values and use the corresponding mixed integer programs to generate rank 1 split cuts our approach was implemented in the coin or framework using cplex 9 0 as a general purpose mip solver we report our computational results on well known benchmark instances from miplib 3 0 and several classes of structured integer and mixed integer problems our computational results show that rank 1 split cuts close more than 98 of the duality gap on 15 out of 41 mixed integer instances from miplib 3 0 more than 75 of the duality gap can be closed on an additional 10 instances the average gap closed over all 41 instances is 72 78 in the pure integer case rank 1 split cuts close more than 75 of the duality gap on 13 out of 24 instances from miplib 3 0 on average rank 1 split cuts close about 72 of the duality gap on these 24 instances we also report results on several classes of structured problems capacitated versions of warehouse location single source facility location p median fixed charge network flow multi commodity network design with splittable and unsplittable flows and lot sizing the fraction of the integrality gap closed varies for these problem classes between 100 and 67 we also gathered statistics on the average coefficient size absolute value of the disjunctions generated they turn out to be surprisingly small