tight formulations for some simple mixed integer programs and convex objective integer programs we study the polyhedral structure of simple mixed integer sets that generalize the two variable set s z r 1 z 1 s z b these sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs for four such sets we give a complete description by valid inequalities and or an integral extended formulation and we also indicate what constraints can be added without destroying integrality we apply these results to provide tight formulations for certain piecewise linear convex objective integer programs and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot sizing problems

a class of markov type operators which preserve e j j 1 abstract in this paper we construct a class of markov type operators which preserve e j j 1 we give a recurrence relation for the moments of this operator and study the convergence we show the uniform convergence of the derivative of order r 0 of the operator a voronovskaya formula and some inequalities are presented

bernstein operators for extended chebyshev systems abstract let u n c n a b be an extended chebyshev space of dimension n 1 suppose that f 0 u n is strictly positive and f 1 u n has the property that f 1 f 0 is strictly increasing we search for conditions ensuring the existence of points t 0 t n a b and positive coefficients 0 n such that for all f c a b the operator b n c a b u n defined by b n f k 0 n f t k k p n k satisfies b n f 0 f 0 and b n f 1 f 1 here it is assumed that p n k k 0 n is a bernstein basis defined by the property that each p n k has a zero of order k at a and a zero of order n k at b

mixing mir inequalities with two divisible coefficients this article is a polyhedral study of a generalization of the mixing set where two different divisible coefficients are allowed for the integral variables our results generalize earlier work on mixed integer rounding mixing and extensions they also directly apply to applications such as production planning problems involving lower bounds or start ups on production when these are modeled as mixed integer linear programs we define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed integer set we give a characterization of each of the maximal faces of the convex hull as well as a closed form description of its extreme points and rays and show how to separate over this set in mathcal o n log n finally we give several extended formulations of polynomial size and study conditions under which adding certain simple constraints on the integer variables preserves our main result

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

quadratic convergence of approximations by ccc schoenberg operators we generalize to the canonical complete chebyshev splines some properties already known for extended chebyshev and piecewise extended chebyshev splines like marsden identity and greville points also we represent an interesting algorithm which leads to numerically stable expressions for the greville points for ccc splines we show that any ccc spline space provides us with infinite number of operators of the schoenberg type and we give a simple characterization of them after proving few important properties we establish a sufficient condition for quadratic convergence of approximations by ccc schoenberg operators to a given function

relaxations and approximations of chance constraints under finite distributions optimization problems with constraints involving stochastic parameters that are required to be satisfied with a prespecified probability threshold arise in numerous applications such chance constrained optimization problems involve the dual challenges of stochasticity and nonconvexity in the setting of a finite distribution of the stochastic parameters an optimization problem with linear chance constraints can be formulated as a mixed integer linear program milp the natural milp formulation has a weak relaxation bound and is quite difficult to solve in this paper we review some recent results on improving the relaxation bounds and constructing approximate solutions for milp formulations of chance constraints we also discuss a recently introduced bicriteria approximation algorithm for covering type chance constrained problems this algorithm uses a relaxation to construct a solution whose constraint violation risk level may be larger than the pre specified threshold but is within a constant factor of it and whose objective value is also within a constant factor of the true optimal value finally we present some new results that improve on the bicriteria approximation factors in the finite scenario setting and shed light on the effect of strong relaxations on the approximation ratios

three letters to walter feit on group representations and quaternions abstract representations of the quaternion group by 2 2 matrices with coefficients in the ring of integers of an imaginary quadratic field

minimum cost capacity installation for multicommodity network flows consider a directed graphg v a and a set of traffic demands to be shipped between pairs of nodes inv capacity has to be installed on the edges of this graph in integer multiples of a base unit so that traffic can be routed in this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs we study two different approaches for solving problems of this type the first one is based on the idea of metric inequalities see onaga and kakusho on feasibility conditions of multicommodity flows in networks ieee transactions on circuit theory ct 18 4 1971 425 429 and uses a formulation with only a variables the second uses an aggregated multicommodity flow formulation and has v a variables we first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on three nodes next we explain our solution methods for both of the approaches in detail and present computational results our computational experience shows that the two formulations are comparable and yield effective algorithms for solving real life problems 1998 the mathematical programming society inc published by elsevier science b v

the 0 1 knapsack problem with a single continuous variable constraints arising in practice often contain many 0 1 variables and one or a small number of continuous variables existing knapsack separation routines cannot be used on such constraints here we study such constraint sets and derive valid inequalities that can be used as cats for such sets as well for more general mixed 0 1 constraints 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 are flows into and out of a single node with binary variable lower and upper bounds on each are 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

note on a korovkin type theorem abstract a new korovkin type theorem and its converse theorem are established we compare our direct result with the korovkin type theorem given by wang 15 as applications we obtain quantitative estimates for q bernstein type operators which preserve the functions e 0 x 1 and e j x x j

mixing mixed integer inequalities mixed integer rounding mir inequalities play a central role in the development of strong cutting planes for mixed integer programs in this paper we investigate how known mir inequalities can be combined in order to generate new strong valid inequalities given a mixed integer region s and a collection of valid base mixed integer inequalities we develop a procedure for generating new valid inequalities for s the starting point of our procedure is to consider the mir inequalities related with the base inequalities for any subset of these mir inequalities we generate two new inequalities by combining or mixing them we show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed integer region associated with the base inequalities we discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed integer programming problems in particular we present examples for production planning capacitated facility location capacitated network design and multiple knapsack problems we also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs finally we study some extensions of this mixing procedure

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

relaxations for two level multi item lot sizing problems we consider several variants of the two level lot sizing problem with one item at the upper level facing dependent demand and multiple items or clients at the lower level facing independent demands we first show that under a natural cost assumption it is sufficient to optimize over a stock dominant relaxation we further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables we consider several variants uncapacitated at both levels with or without start up costs uncapacitated at the upper level and constant capacity at the lower level constant capacity at both levels we finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products

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

lot sizing with constant batches formulation and valid inequalities we consider the classical lot sizing problem with constant production capacities lcc and a variant in which the capacity in each period is an integer multiple of some basic batch size lcb we first show that the classical dynamic program for lcc simplifies for lcb leading to an on2 min n c algorithm where n is the number of periods and c the batch size using this new algorithm we show how to formulate both problems as linear programs with on3 variables and constraints a class of so called k l s i inequalities are described for lcb which capture both the dynamic nature of the problem as well as the capacity aspects for lcb we prove that these inequalities are the only facet defining inequalities of a certain form for lcc we show that these inequalities include all the known classes of valid inequalities finally we discuss several open questions and possible extensions

on a class of weak tchebycheff systems in this paper we study the approximation power the existence of a normalized b basis and the structure of a degree raising process for spaces of the form requiring suitable assumptions on the functions u and v the results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of cagd

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

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

on capacitated network design cut set polyhedra this paper provides an analysis of capacitated network design cut set polyhedra we give a complete linear description of the cut set polyhedron of the single commodity single facility capacitated network design problem then we extend the analysis to single commodity multifacility and multicommodity multifacility capacitated network design problems valid inequalities described here are applicable to directed network design problems with any number of facility types and any level of capacities we report results from a computational study done for testing the effectiveness of the new inequalities

on the facets of the mixed integer knapsack polyhedron 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

hilbert bases and the facets of special knapsack polytopes we present a linear description of the 0 1 knapsack polytope for the special case when the number of different weights of the items is two the inequalities needed for this description involve elements of the hilbert basis of the cone of exchange vectors

shape preserving properties of generalized bernstein operators on extended chebyshev spaces we study the existence and shape preserving properties of a generalized bernstein operator b n fixing a strictly positive function f 0 and a second function f 1 such that f 1 f 0 is strictly increasing within the framework of extended chebyshev spaces u n the first main result gives an inductive criterion for existence suppose there exists a bernstein operator b n c a b u n with strictly increasing nodes fixing f 0 f 1 in u n if u n subset u n 1 and u n 1 has a non negative bernstein basis then there exists a bernstein operator b n 1 c a b u n 1 with strictly increasing nodes fixing f 0 and f 1 in particular if f 0 f 1 f n is a basis of u n such that the linear span of f 0 f k is an extended chebyshev space over a b for each k 0 n then there exists a bernstein operator b n with increasing nodes fixing f 0 and f 1 the second main result says that under the above assumptions the following inequalities hold b n f geq b n 1 f geq f furthermore b nf is f0 f1 convex for all f0 f1 convex functions f in c a b

chebyshev blossoming in muntz spaces toward shaping with young diagrams the notion of a blossom in extended chebyshev spaces offers adequate generalizations and extra utilities to the tools for free form design schemes unfortunately such advantages are often overshadowed by the complexity of the resulting algorithms in this work we show that for the case of muntz spaces with integer exponents the notion of a chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of schur functions we express the blossom and the pseudo affinity property in muntz spaces in terms of schur functions we derive an explicit expression for the chebyshev bernstein basis via an inductive argument on nested muntz spaces we also reveal a simple algorithm for dimension elevation free form design schemes in muntz spaces with young diagrams as shape parameters are discussed

mixing polyhedra with two non divisible coefficients we consider the mixed integer set x s x y in mathbb r times mathbb z n times mathbb z m s a 1 x j geq b j forall j in n 1 s a 2 y j geq d j j in n 2 where n 1 1 ldots n n 2 1 ldots m and a 1 a 2 in mathbb z setminus 0 this set may arise in a relaxation of mixed integer problems such as lot sizing problems we decompose x into a small number of subsets whose convex hull description is trivial the convex hull of x is equal to the closure of the convex hull of the union of those polyhedra using a projection theorem from balas discret appl math 89 3 44 1998 we obtain a compact characterization of the facets of the convex hull of x then by studying the projection cone we characterize all the facet defining inequalities of the convex hull of x in the space of the original variables each of those inequalities is either a mixed mir inequality gunluk and pochet in math programm 90 429 457 2001 or it is based on a directed cycle on a special bipartite graph when a 1 and a 2 are relative prime the convex hull of x is described by the mixed mir 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 optimal stability of bases of univariate functions this paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation the only bases considered are those whose elements have no sign changes among these an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval a uniqueness result and many examples of such optimally stable bases are also provided

polyhedra for lot sizing with wagner whitin costs we examine the single item lot sizing problem with wagner whitin costs over ann period horizon i e pt ht pt 1 fort 1 n 1 wherept ht are the unit production and storage costs in periodt respectively so it always pays to produce as late as possible

even unimodular lattices associated with the weil representation of the finite symplectic group soit q une puissance d un nombre premier impair p et soit g le groupe symplectique sp 2n q quand n est pair et q p est un nombre premier avec p 3 mod 4 il existe un treillis integral rationnel g invariant l dans m qui supporte une forme unimodulaire definie positive symetrique paire m est un qg module et q le corps des rationnels

finite groups over arithmetical rings and globally irreducible representations abstract given the ring of integersrof an algebraic number fieldk for which natural numbernis there a finite groupg gl n r such thatrg ther span ofg coincides withm n r the ring of n n matrices overr giveng gl n r we show thatrg m n r if and only if the brauer reduction ofgmodulo every prime is absolutely irreducible in addition the question above is fully answered ifnis an odd prime

mixing sets linked by bidirected paths recently there has been considerable research on simple mixed integer sets called mixing sets and closely related sets arising in uncapacitated and constant capacity lot sizing this in turn has led to study of more general sets called network dual sets for which it is possible to derive extended formulations whose projection gives the convex hull of the network dual set unfortunately this formulation cannot be used in general to optimize in polynomial time furthermore the inequalities defining the convex hull of a network dual set in the original space of variables are known only for some special cases here we study two new cases in which the continuous variables of the network dual set are linked by a bidirected path in the first case which is motivated by lot sizing problems with lost sales we provide a description of the convex hull as the intersection of the convex hulls of 2 n mixing sets where n is the number of continuous variables of the set however optimization is polynomial as only n 1 of the sets are required for any given objective function in the second case generalizing single arc flow sets we describe again the convex hull as the intersection of an exponential number of mixing sets and also give a combinatorial polynomial time separation algorithm

bernstein type operators which preserve polynomials abstract in this paper we present the sequence of linear bernstein type operators defined for f c 0 1 by b n f 1 b n being the classical bernstein operators and being any function that is continuously differentiable times on 0 1 such that 0 0 1 1 and x 0 for x 0 1 we investigate its shape preserving and convergence properties as well as its asymptotic behavior and saturation moreover these operators and others of king type are compared with each other and with b n we present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties which converge to the identity which in a certain sense improve b n in approximating a number of increasing functions and which apart from the constant functions fix suitable polynomials of a prescribed degree the notion of convexity with respect to plays an important role

schur indices and schur groups w x schur groups were introduced in 8 as a certain kind of smallest groups admitting characters with nontrivial schur index indeed every finite group g having an irreducible character x with m x 1 involves such schur groups they occur as terminal steps by repeated application of the brauer witt theorem over local fields of characteristic zero there is a connection with the schur subgroup s k of the brauer group br k of such a field k this s k has been studied in detail by w x w x yamada 12 and quite recently by adem and milgram 1 from the character theoretic point of view it appears to be of interest to give explicit generators for s k in terms of schur groups see section 6 in the course w x of this investigation the authors noticed that there must be a gap in 8 concerning the exceptional dyadic schur groups yielding nontrivial 2 lo cal index it turned out that the definition of the schur groups of type qd q was too restrictive the correct definition will be given in section 5 below we take the opportunity to discuss these terminal q reduc2 tions once again and the class of 2 groups involved theorem 4 and 5 the schur groups admit faithful irreducible characters which are induced from linear characters of cyclic normal subgroups for the sake of clarity we begin our present work by making some general comments on schur indices in group extensions this should be compared with turull w x 11 our approach is perhaps more elementary and we just demonstrate w x how to use the results on stable clifford theory 7 more details may be w x found in the recent thesis 5 of the first named author

modeling and solving the two facility capacitated network loading problem this paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry in this problem given point to point communication demand in a network must be met by installing loading capacitated facilities on the arcs loading a facility incurs an arc specific and facility dependent cost this paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost we consider two approaches for solving the underlying mixed integer program a lagrangian relaxation strategy and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem we show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the lagrangian relaxation bound our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation from an average of 25 to an average of 8 these results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

group representations and lattices this paper began as an attempt to understand the euclidean lattices that were recently constructed using the theory of elliptic curves by elkies e and shioda shl sh2 from the point of view of group representations the main idea appears in a note of thompson th2 if one makes strong irreducibility hypotheses on a rational representation v of a finite group g then the g stable euclidean lattices in v are severely restricted unfortunately these hypotheses are rarely satisfied when v is absolutely irreducible over q but there are many examples where the ring endg v is an imaginary quadratic field or a definite quaternion algebra these representations allow us to construct some of the mordell weil lattices considered by elkies and shioda as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves in 1 we discuss lattices and hermitian forms on y7 and in 2 4 the strong irreducibility hypotheses we wish to make in 5 we show how our hypotheses imply the existence of a finite number up to isomorphism of euclidean z g lattices l in v with endg l a maximal order in endg v we give some examples with dim v 8 in 6 and in 7 9 discuss the invariants of l such as the dual lattice and theta function the rest of the paper is devoted to examples in most g is a finite group of lie type and v is obtained as an irreducible summand of the weil representation of g some of the representation theoretic problems left open by this paper are to find all examples of pairs g v satisfying the strong irreducibility hypotheses of 2 4 and to determine the invariants shortest nonzero vector theta function thompson series of the g lattices l so effortlessly constructed inside v

lower bounds in lot sizing models a polyhedral study variable lower bounds in mixed integer programs are constraints with the general form x ly where xis a continuous variable and y is a binary or an integer variable this type of constraints is present in some lot sizing models where x represents the amount of some article produced by a machine and y 0 1 indicates whether the machine is set up for that articley 1 or not y 0 in these models production below some level is not allowed in order to make full use of resources in this paper we study from the polyhedral viewpoint the mixed integer models whose feasible set includes an additive constraint xj d and constraints lyj xj kyj where xj are continuous variables yj are integer or binary variables and k is a large constant we derive families of strong valid inequalities and show they are sufficient to describe completely the convex hulls of the sets of feasible solutions moreover we develop polynomial algorithms to solve the separation problem associated to each of the families obtained finally we incorporate the inequalities obtained in a cutting plane algorithm and test their ability to solve some multi item lot sizing problems

complete asymptotic expansions related to conjecture on a voronovskaja type theorem abstract we provide complete asymptotic expansions for some sequences of bernstein type and meyer konig and zeller type operators preserving the monomials e 0 and e j j 1 in particular this answers a conjecture related to a voronovskaja type theorem

lifting two integer knapsack inequalities 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

convergence of rational bernstein operators abstract in this paper we discuss convergence properties and error estimates of rational bernstein operators introduced by pi ul and sablonniere it is shown that the rational bernstein operators converge to the identity operator if and only if the maximal difference between two consecutive nodes is converging to zero further a voronovskaja theorem is given based on the explicit computation of higher order moments for the rational bernstein operator

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

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

capacitated facility location valid inequalities and facets we examine the polyhedral structure of the convex hull of feasible solutions of the capacitated facility location problem in particular we derive necessary and sufficient conditions for a family of effective capacity inequalities to be facet defining and further results on a more general family called submodular inequalities