solving multi item lot sizing problems with an mip solver using classification and reformulation based on research on the polyhedral structure of lot sizing models over the last 20 years we claim that there is a nontrivial fraction of practical lot sizing problems that can now be solved by nonspecialists just by taking an appropriate a priori reformulation of the problem and then feeding the resulting formulation into a commercial mixed integer programming solver this claim uses the fact that many multi item problems decompose naturally into a set of single item problems with linking constraints and that there is now a large body of knowledge about single item problems to put this knowledge to use we propose a classification of lot sizing problems in large part single item and then indicate in a set of tables what is known about a particular problem class and how useful it might be specifically we indicate for each class i whether a tight extended formulation is known and its size ii whether one or more families of valid inequalities are known defining the convex hull of solutions and the complexity of the corresponding separation algorithms and iii the complexity of the corresponding optimization algorithms which would be useful if a column generation or lagrangian relaxation approach was envisaged three distinct multi item lot sizing instances are then presented to demonstrate the approach and comparative computational results are presented finally we also use the classification to point out what appear to be some of the important open questions and challenges

algorithms for single item lot sizing problems with constant batch size the main result of this paper is an o n3 algorithm for the single item lot sizing problem with constant batch size and backlogging we consider a general number of installable batches i e in each time period t we may produce up to mt batches where the mt are given and time dependent this generalizes earlier results as we consider backlogging and a general number of maximum batches we also give faster algorithms for three special cases of this general problem when backlogging is not allowed and the costs satisfy the wagner whitin property the problem is solvable in o n2 log n time when the production in each period is required to be either zero or equal to the installed capacity it is possible to solve the problem with and without backlogging in o n2 and o n log n time respectively

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

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

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

dynamical systems disjoint from any minimal system furstenberg showed that if two topological systems x t and y s are disjoint then one of them say y s is minimal when y s is nontrivial we prove that x t must have dense recurrent points and there are countably many maximal transitive subsystems of x t such that their union is dense and each of them is disjoint from y s showing that a weakly mixing system with dense periodic points is in m the collection of all systems disjoint from any minimal system furstenberg asked the question to characterize the systems in m we show that a weakly mixing system with dense regular minimal points is in m and each system in m has dense minimal points and it is weakly mixing if it is transitive transitive systems in m and having no periodic points are constructed moreover we show that there is a distal system in m recently weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic we construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing

facets and algorithms for capacitated lot sizing the dynamic economic lot sizing model which lies at the core of numerous production planning applications is one of the most highly studied models in all of operations research and yet capacitated multi item versions of this problem remain computationally elusive we study the polyhedral structure of an integer programming formulation of a single item capacitated version of this problem and use these results to develop solution methods for multi item applications in particular we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane branch and bound procedure moreover our results show that in many instances adding certain of these inequalities a priori to the problem formulation and avoiding the generation of cutting planes can be equally effective

compact formulations as a union of polyhedra we explore one method for finding the convex hull of certain mixed integer sets the approach is to break up the original set into a small number of subsets find a compact polyhedral description of the convex hull of each subset and then take the convex hull of the union of these polyhedra the resulting extended formulation is then compact its projection is the convex hull of the original set and optimization over the mixed integer set is reduced to solving a linear program over the extended formulation the approach is demonstrated on three different sets a continuous mixing set with an upper bound and a mixing set with two divisible capacities both arising in lot sizing and a single node flow model with divisible capacities that arises as a subproblem in network design

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

tight second stage formulations in two stage stochastic mixed integer programs we study two stage stochastic mixed integer programs tss mips with integer variables in the second stage we show that under suitable conditions the second stage mips can be convexified by adding parametric cuts a priori as special cases we extend the results of miller and wolsey math program 98 2003 pp 73 88 to tss mips furthermore we consider second stage programs that are generalizations of the well known mixing and continuous mixing set or certain piecewise linear convex objective integer programs these results allow us to relax the integrality restrictions on the second stage integer variables without effecting the integrality of the optimal solution of the tss mip we also use four variants of the two stage stochastic capacitated lot sizing problems as test problems for computational experiments and present tight second stage formulations for these problems our computational results show that adding parametric inequalities that a priori convexify the second stage formulation si

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

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

a linear programming approach for linear programs with probabilistic constraints we study a class of mixed integer programs for solving linear programs with joint probabilistic constraints from random right hand side vectors with finite distributions we present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs we provide optimality gaps for our heuristic solutions via the linear programming relaxation of the extended mixed integer formulation of luedtke et al 2010 13 as well as via lower bounds produced by their cutting plane method while we demonstrate through an extensive computational study the effectiveness and scalability of our heuristics we also prove that the theoretical worst case solution quality for these algorithms is arbitrarily far from optimal our computational study compares our heuristics against both the extended mixed integer programming formulation and the cutting plane method of luedtke et al 2010 13 our heuristics efficiently and consistently produce solutions with small optimality gaps while for larger instances the extended formulation becomes intractable and the optimality gaps from the cutting plane method increase to over 5

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

anosov flows surface groups and curves in projective space note that in 10 w goldman gives a complete description of these connected components in the case of finite covers of psl 2 r in the case of psl 2 r two homeomorphic components called teichmuller spaces play a central role these two components are well known to be homeomorphic to a ball of dimension 6g 6 n hitchin generalises this situation to psl n r indeed one of these components when n is odd and two when n is even has a very simple topology let us define an n fuchsian representation to be a representation which can be written as 0 where 0 is a cocompact representation with values in psl 2 r and is the irreducible representation of psl 2 r in psl n r we denote by reph 1 s psl n r a connected component that contains fuchsian representations and call it a hitchin component in fact when n is odd there is one hitchin component and when n is even two isomorphic ones n hitchin proves in 17

economic lot sizing an o n log n algorithm that runs in linear time in the wagner whitin case we consider the n period economic lot sizing problem where the cost coefficients are not restricted in sign in their seminal paper h m wagner and t m whitin proposed an o n2 algorithm for the special case of this problem where the marginal production costs are equal in all periods and the unit holding costs are nonnegative it is well known that their approach can also be used to solve the general problem without affecting the complexity of the algorithm in this paper we present an algorithm to solve the economic lot sizing problem in o n log n time and we show how the wagner whitin case can even be solved in linear time our algorithm can easily be explained by a geometrical interpretation and the time bounds are obtained without the use of any complicated data structure furthermore we show how wagner and whitin s and our algorithm are related to algorithms that solve the dual of the simple plant location formulation of the economic lot sizing problem

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

on the capacitated lot sizing and continuous 0 1 knapsack polyhedra abstract we consider the single item capacitated lot sizing problem a well known production planning model that often arises in practical applications and derive new classes of valid inequalities for it we first derive new easily computable valid inequalities for the continuous 0 1 knapsack problem which has been introduced recently and has been shown to provide a useful relaxation of mixed 0 1 integer programs we then show that by studying an appropriate continuous 0 1 knapsack relaxation of the capacitated lot sizing problem it is possible not only to derive new classes of valid inequalities for the capacitated lot sizing problem but also to derive and or strengthen several known classes

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

single item lot sizing with non decreasing capacities we consider the single item lot sizing problem with capacities that are non decreasing over time when the cost function is i non speculative or wagner whitin for instance constant unit production costs and non negative unit holding costs and ii the production set up costs are non increasing over time it is known that the minimum cost lot sizing problem is polynomially solvable using dynamic programming when the capacities are non decreasing we derive a compact mixed integer programming reformulation whose linear programming relaxation solves the lot sizing problem to optimality when the objective function satisfies i and ii the formulation is based on mixing set relaxations and reduces to the known convex hull of solutions when the capacities are constant over time we illustrate the use and potential effectiveness of this improved lp formulation on a few test instances including instances with and without wagner whitin costs and with both non decreasing and arbitrary capacities over time

strong formulations for mixed integer programs valid inequalities and extended formulations we examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the easy and hard sides of the complexity frontier most progress has been made in studying sets arising from knapsack and single node flow sets and a variety of sets motivated by different lot sizing models we conclude by citing briefly some of the more intriguing new avenues of research

solving multi item lot sizing problems using strong cutting planes we consider two classes of multi item lot sizing problems the first is a class of single stage problems involving joint machine capacity constraints and or start up costs and the second is a class of multistage problems with general product structure the problems are solved as mixed integer programs based on i an appropriate choice of the initial problem formulation and ii the addition of cuts which are generated automatically by a mathematical programming system mpsarx our results extend and complement those of karmarkar and schrage 1985 afentakis and gavish 1986 eppen and martin 1987 and van roy and wolsey 1987 a major advantage of this approach is its robustness or flexibility by using just a matrix generator and a mathematical programming system with automatic cut generation routines we can formulate and solve model variants without incurring the costs of adapting an algorithm

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

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

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

a study of the lot sizing polytope the lot sizing polytope is a fundamental structure contained in many practical production planning problems here we study this polytope and identify facet defining inequalities that cut off all fractional extreme points of its linear programming relaxation as well as liftings from those facets we give a polynomial time combinatorial separation algorithm for the inequalities when capacities are constant we also report computational experiments on solving the lot sizing problem with varying cost and capacity characteristics

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

computational complexity of the capacitated lot size problem in this paper we study the computational complexity of the capacitated lot size problem with a particular cost structure that is likely to be used in practical settings for the single item case new properties are introduced classes of problems solvable by polynomial time algorithms are identified and efficient solution procedures are given we show that special classes are np hard and that the problem with two items and independent setups is np hard under conditions similar to those where the single item problem is easy topics for further research are discussed in the last section

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

mapping the geometry of the e6 group in this paper we present a construction for the compact form of the exceptional lie group e sub 6 by exponentiating the corresponding lie algebra e sub 6 which we realize as the sum of f sub 4 the derivations of the exceptional jordan algebra j sub 3 of dimension 3 with octonionic entries and the right multiplication by the elements of j sub 3 with vanishing trace our parameterization is a generalization of the euler angles for su 2 and it is based on the fibration of e sub 6 via a f sub 4 subgroup as the fiber it makes use of a similar construction we have performed in a previous article for f sub 4 an interesting first application of these results lies in the fact that we are able to determine an explicit expression for the haar invariant measure on the e sub 6 group manifold

modelling practical lot sizing problems as mixed integer programs in spite of the remarkable improvements in the quality of general purpose mixed integer programming software the effective solution of a variety of lot sizing problems depends crucially on the development of tight formulations for the special problem features occurring in practice after reviewing some of the basic preprocessing techniques for handling safety stocks and multilevel problems we discuss a variety of aspects arising particularly in small and large bucket time period models such as start ups changeovers minimum batch sizes choice of one or two set ups per period etc a set of applications is described that contains one or more of these special features and some indicative computational results are presented finally to show another technique that is useful a slightly different supply chain application is presented for which the a priori addition of some simple mixed integer inequalities based on aggregation leads to important improvements in the results

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

convex hull representation of the deterministic bipartite network interdiction problem we consider a version of the stochastic network interdiction problem modeled by morton et al iie trans 39 3 14 2007 in which an interdictor attempts to minimize a potential smuggler s chance of evasion via discrete deployment of sensors on arcs in a bipartite network the smuggler reacts to sensor deployments by solving a maximum reliability path problem on the resulting network in this paper we develop the minimal convex hull representation for the polytope linking the interdictor s decision variables with the smuggler s for the case in which the smuggler s origin and destination are known and interdictions are cardinality constrained in the process we propose an exponential class of easily separable inequalities that generalize all of those developed so far for the bipartite version of this problem we show how these cuts may be employed in a cutting plane fashion when solving the more difficult problem in which the smuggler s origin and destination are stochastic and argue that some instances of the stochastic model have facets corresponding to the solution of np hard problems our computational results show that the cutting planes developed in this paper may strengthen the linear programming relaxation of the stochastic model by as much as 25

dynamic knapsack sets and capacitated lot sizing a dynamic knapsack set is a natural generalization of the 0 1 knapsack set with a continuous variable studied recently for dynamic knapsack sets a large family of facet defining inequalities called dynamic knapsack inequalities are derived by fixing variables to one and then lifting surprisingly such inequalities have the simultaneous lifting property and for small instances provide a significant proportion of all the facet defining inequalities we then consider single item capacitated lot sizing problems and propose the joint study of three related sets the first models the discrete lotsizing problem the second the continuous lot sizing problem with wagnerwhitin costs and the third the continuous lot sizing problem with arbitrary costs the first set that arises is precisely a dynamic knapsack set the second an intersection of dynamic knapsack sets and the unrestricted problem can be viewed as both a relaxation and a restriction of the second it follows that the dynamic knapsack inequalities and their generalizations provide strong valid inequalities for all three sets some limited computation results are reported as an initial test of the effectiveness of these inequalities on capacitated lot sizing problems

a polyhedral study of multiechelon lot sizing with intermediate demands in this paper we study a multiechelon uncapacitated lot sizing problem in series m uls where the output of the intermediate echelons has its own external demand and is also an input to the next echelon we propose a polynomial time dynamic programming algorithm which gives a tight compact extended formulation for the two echelon case 2 uls next we present a family of valid inequalities for m uls show its strength and give a polynomial time separation algorithm we establish a hierarchy between the alternative formulations for 2 uls in particular we show that our valid inequalities can be obtained from the projection of the multicommodity formulation our computational results show that this extended formulation is very effective in solving our uncapacitated multi item two echelon test problems in addition for capacitated multi item multiechelon problems we demonstrate the effectiveness of a branch and cut algorithm using the proposed inequalities

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

an integer programming approach for linear programs with probabilistic constraints linear programs with joint probabilistic constraints pclp are difficult to solve because the feasible region is not convex we consider a special case of pclp in which only the right hand side is random and this random vector has a finite distribution we give a mixed integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint we obtain two strengthened formulations as a byproduct of this analysis we obtain new results for the previously studied mixing set subject to an additional knapsack inequality we present computational results which indicate that by using our strengthened formulations instances that are considerably larger than have been considered before can be solved to optimality

a polyhedral study on chance constrained program with random right hand side the essential structure of the mixed integer programming formulation for chance constrained program ccp is the intersection of multiple mixing sets with a 0 1 knapsack to improve our computational capacity on ccp an underlying substructure the single mixing set with a 0 1 knapsack has received substantial attentions recently in this study we consider a ccp problem with stochastic right hand side under a finite discrete distribution we first present a family of strong inequalities that subsumes known facet defining ones for that single mixing set due to the flexibility of our generalized inequalities we develop a new separation heuristic that has a complexity much less than existing one and guarantees generated cutting planes are facet defining for the polyhedron of ccp then we study lifting and superadditive lifting on knapsack cover inequalities and provide an implementable procedure on deriving another family of strong inequalities for the single mixing set finally different from the traditional approach that aggregates original constraints to investigate polyhedral implications due to their interactions we propose a novel blending procedure that produces strong valid inequalities for ccp by integrating those derived from individual mixing sets we show that under certain conditions they are the first type of facet defining inequalities describing intersection of multiple mixing sets and design an efficient separation heuristic for implementation in the computational experiments we perform a systematic study and illustrate the efficiency of the proposed inequalities on solving chance constrained static probabilistic lot sizing problems

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

a simple forward algorithm to solve general dynamic lot sizing models with n periods in 0n log n or 0n time this paper is concerned with the general dynamic lot size model or generalized wagner whitin model let n denote the number of periods into which the planning horizon is divided we describe a simple forward algorithm which solves the general model in 0n log n time and 0n space as opposed to the well known shortest path algorithm advocated over the last 30 years with 0n2 time a linear i e 0n time and space algorithm is obtained for two important special cases a models without speculative motives for carrying stock i e where in each interval of time the per unit order cost increases by less than the cost of carrying a unit in stock b models with nondecreasing setup costs we also derive conditions for the existence of monotone optimal policies and relate these to known planning horizon and other results from the literature

on the mixing set with a knapsack constraint we study a substructure appearing in mixed integer programming reformulations of chance constrained programs with stochastic right hand sides over a finite discrete distribution which we call the mixing set with a knapsack constraint recently luedtke et al math program 122 2 247 272 2010 and kucukyavuz math program 132 1 31 56 2012 studied valid inequalities for such sets however most of their results were focused on the equal probabilities case when the knapsack constraint reduces to a cardinality constraint in this paper we focus on the general probabilities case general knapsack constraint we characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the results previously derived for the equal probabilities case our results allow for a deep understanding of the relationship that the set under consideration has with the knapsack polytope moreover they allow us to establish benchmarks that can be used to identify when a relaxation will be useful for the considered types of reformulations of chance constrained programs

a note on exceptional groups and reidemeister torsion let be a closed orientable surface of genus at least 2 and g be one of the exceptional groups g2 f4 and e6 the present article considers the set rep g of g valued representations from the fundamental group 1 of the surface to the exceptional group g it proves that for such representations the notion of reidemeister torsion is well defined it also establishes a formula for computing reidemeister torsion of such representations in terms of the well known symplectic structure on rep g namely the atiyah bott goldman symplectic form for the lie group g moreover it applies to g valued hitchin representations

network formulations of mixed integer programs we consider mixed integer sets described by system of linear inequalities in which the constraint matrix a is totally unimodular the right hand side is arbitrary vector and a subset of the variables is required to be integer we show that the problem of checking nonemptiness of a set of this type is np complete even in the case in which the linear system describes mixed integer network flows with half integral requirement on the nodes this is in contrast to the case in which a is totally unimodular and contains at most two nonzeros per row in this case we provide an extended formulation for the convex hull of solutions whose constraint matrix is a dual network matrix with an integral right hand side vector the size of this formulation depends on the number of distinct fractional parts taken by the continuous variables in the extreme points of the convex hull of the given set when this number is polynomial in the dimension of a the extended formulation is of polynomial size if in addition the corresponding list of fractional parts can be computed efficiently then our result provides a polynomial algorithm for the optimization problem over these sets we show that there are instances for which this list is of exponential size and we also give conditions under which it is short and can be efficiently computed finally we show that these results provide a unified framework leading to polynomial size extended formulations for several generalizations of mixing sets and lot sizing sets studied in the last few years

on mixing sets arising in chance constrained programming the mixing set with a knapsack constraint arises in deterministic equivalent of chance constrained programming problems with finite discrete distributions we first consider the case that the chance constrained program has equal probabilities for each scenario we study the resulting mixing set with a cardinality constraint and propose facet defining inequalities that subsume known explicit inequalities for this set we extend these inequalities to obtain valid inequalities for the mixing set with a knapsack constraint in addition we propose a compact extended reformulation with polynomial number of variables and constraints that characterizes a linear programming equivalent of a single chance constraint with equal scenario probabilities we introduce a blending procedure to find valid inequalities for intersection of multiple mixing sets we propose a polynomial size extended formulation for the intersection of multiple mixing sets with a knapsack constraint that is stronger than the original mixing formulation we also give a compact extended linear program for the intersection of multiple mixing sets and a cardinality constraint for a special case we illustrate the effectiveness of the proposed inequalities in our computational experiments with probabilistic lot sizing problems

nilsequences and a structure theorem for topological dynamical systems abstract we characterize inverse limits of nilsystems in topological dynamics via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems we provide two applications of the structure the first is to nilsequences which have played an important role in recent developments in ergodic theory and additive combinatorics we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally meaning on finite intervals the second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system we show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1

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

valid inequalities for separable concave constraints with indicator variables we study valid inequalities for optimization models that contain both binary indicator variables and separable concave constraints these models reduce to a mixed integer linear program milp when the concave constraints are ignored or to a nonconvex global optimization problem when the binary restrictions are ignored in algorithms designed to solve these problems to global optimality cutting planes to strengthen the relaxation are traditionally obtained using valid inequalities for the milp only we propose a technique to obtain valid inequalities that are based on both the milp constraints and the concave constraints we begin by characterizing the convex hull of a four dimensional set consisting of a single binary indicator variable a single concave constraint and two linear inequalities using this analysis we demonstrate how valid inequalities for the single node flow set and for the lot sizing polyhedron can be tilted to give valid inequalities that also account for separable concave functions of the arc flows we present computational results demonstrating the utility of the new inequalities for nonlinear transportation problems and for lot sizing problems with concave costs to our knowledge this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed integer nonlinear programs

a heuristic approach for big bucket multi level production planning problems multi level production planning problems in which multiple items compete for the same resources frequently occur in practice yet remain daunting in their difficulty to solve in this paper we propose a heuristic framework that can generate high quality feasible solutions quickly for various kinds of lot sizing problems in addition unlike many other heuristics it generates high quality lower bounds using strong formulations and its simple scheme allows it to be easily implemented in the xpress mosel modeling language extensive computational results from widely used test sets that include a variety of problems demonstrate the efficiency of the heuristic particularly for challenging problems

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

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

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

a cutting plane approach to capacitated lot sizing with start up costs we consider a mixed integer model for multi item single machine production planning incorporating both start up costs and machine capacity the single item version of this model is studied from the polyhedral point of view and several families of valid inequalities are derived for some of these inequalities we give necessary and sufficient facet inducing conditions and efficient separation algorithms we use these inequalities in a cutting plane branch and bound procedure a set of real life based problems with 5 items and up to 36 periods is solved to optimality

n n step cycle inequalities facets for continuous multi mixing set and strong cuts for multi module capacitated lot sizing problem in this paper we introduce a generalization of the continuous mixing set which we refer to as the continuous multi mixing set this set is closely related to the feasible set of the multi module capacitated lot sizing mml problem with out backlogging we present a family of valid inequalities for the continuous multi mixing set and identify conditions under which they are facet defining the cycle inequalities n n step mir inequalities and mixed n n step mir inequalities are special cases of these inequalities we also present an exact separation algorithm for our inequalities we then use these inequalities to generate valid inequalities for mml with out backlogging our computational results show that our cuts are very effective in solving the mml instances resulting in substantial reduction in the integrality gap number of nodes and total solution time

regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence abstract inverse limits of nilsystems in topologically dynamical systems were characterized by host kra and maass recently namely for each d n a certain generalization of the regionally proximal relation was introduced and for a distal minimal system it was shown that such a relation is an equivalence one which determines the maximal d step nilfactor one of the main results in this article is to show that the above results hold for a general minimal system a combinatorial consequence is also deduced which is the topological correspondence of the result obtained by host and kra for positive upper banach density subsets using ergodic methods

higher order regionally proximal equivalence relations for general minimal group actions abstract we introduce higher order regionally proximal relations suitable for an arbitrary acting group for minimal abelian group actions these relations coincide with the ones introduced by host kra and maass our main result is that these relations are equivalence relations whenever the action is minimal this was known for abelian actions by a result of shao and ye we also show that these relations lift through extensions between minimal systems answering a question by tao given a minimal system we prove that the regionally proximal equivalence relation of order d corresponds to the maximal dynamical antolin camarena szegedy nilspace factor of order at most d in particular the regionally proximal equivalence relation of order one corresponds to the maximal abelian group factor finally by using a result of gutman manners and varju under some restrictions on the acting group it follows that the regionally proximal equivalence relation of order d corresponds to the maximal pronilfactor of order at most d a factor which is an inverse limit of nilsystems of order at most d

a branch and cut algorithm for the stochastic uncapacitated lot sizing problem this paper addresses a multi stage stochastic integer programming formulation of the uncapacitated lot sizing problem under uncertainty we show that the classical l s inequalities for the deterministic lot sizing polytope are also valid for the stochastic lot sizing polytope we then extend the l s inequalities to a general class of valid inequalities called the inequalities and we establish necessary and sufficient conditions which guarantee that the inequalities are facet defining a separation heuristic for inequalities is developed and incorporated into a branch and cut algorithm a computational study verifies the usefulness of the inequalities as cuts

the structure theory of nilspaces iii inverse limit representations and topological dynamics abstract this paper forms the third part of a series by the authors 12 11 concerning the structure theory of nilspaces a nilspace is a compact space x together with closed collections of cubes c n x x 2 n n 1 2 satisfying some natural axioms our goal is to extend the structure theory of nilspaces obtained by antolin camarena and szegedy and to provide new proofs our main result is that under the technical assumption that c n x is a connected space for all n then x is isomorphic in a strong sense to an inverse limit of nilmanifolds this is a direct and slight generalization of the main result of antolin camarena and szegedy we also apply our methods to obtain structure theorems in the setting of topological dynamics specifically if h is a group subject to very mild topological assumptions and h x is a minimal dynamical system then we give a simple characterization of the maximal pronilfactor of x this generalizes the case h z which is a theorem of host kra and maass although even in that case we give a significantly different proof

lot sizing with inventory bounds and fixed costs polyhedral study and computation we investigate the polyhedral structure of the lot sizing problem with inventory bounds we consider two models one with linear cost on inventory the other with linear and fixed costs on inventory for both models we identify facet defining inequalities that make use of the inventory bounds explicitly and give exact separation algorithms we also describe a linear programming formulation of the problem when the order and inventory costs satisfy the wagner whitin nonspeculative property we present computational experiments that show the effectiveness of the results in tightening the linear programming relaxations of the lot sizing problem with inventory bounds and fixed costs

projecting an extended formulation for mixed integer covers on bipartite graphs we consider the mixed integer version of bipartite vertex cover this is equivalent to a mixed integer network dual model introduced recently that generalizes several mixed integer sets arising in production planning we derive properties of inequalities that are valid for the convex hull of the mixed integer bipartite covers by projecting an extended formulation onto the space of the original variables this permits us to give a complete description of the facet inducing inequalities of the double mixing set and of the continuous mixing set with flows two mixed integer sets that generalize several models studied in the literature

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