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

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

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

capacitated facility location separation algorithms and computational experience we consider the polyhedral approach to solving the capacitated facility location problem the valid inequalities considered are the knapsack cover flow cover effective capacity single depot and combinatorial inequalities the flow cover effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities the separation problem based on the family of submodular inequalities is np hard in general for the well known subclass of flow cover inequalities however we show that if the client set is fixed and if all capacities are equal then the separation problem can be solved in polynomial time for the flow cover inequalities based on an arbitrary client set and general capacities and for the effective capacity and single depot inequalities we develop separation heuristics an important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining the way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities the family of combinatorial inequalities was originally developed for the uncapacitated facility location problem but is also valid for the capacitated problem no computational experience using the combinatorial inequalities has been reported so far here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities we report on computational results from solving 60 medium size problems

a comparison of heuristics and relaxations for the capacitated plant location problem abstract approaches proposed in the literature for the capacitated plant location problem are compared the comparison is based on new theoretical and computational results the main emphasis is on relaxations in particular dominance relations among the various relaxations found in the literature are identified in the computational study the relaxations are compared as a function of various characteristics of the test problems several of these relaxations can be used to generate heuristic feasible solutions that are better than the classical greedy or interchange heuristics both in computing time and in the quality of the solutions found

0 1 2 chva tal gomory cuts given the integer polyhedronp t conv x n ax b wherea m n andb m achvatal gomory cg cut is a valid inequality forp 1 of the type ax b for some such that a n in this paper we study 0 1 2 cg cuts arising for 0 1 2 m we show that the associated separation problem 0 1 2 sep is equivalent to finding a minimum weight member of a binary clutter this implies that 0 1 2 sep is np complete in the general case but polynomially solvable whena is related to the edge path incidence matrix of a tree we show that 0 1 2 sep can be solved in polynomial time for a convenient relaxation of the systemax b this leads to an efficient separation algorithm for a subclass of 0 1 2 cg cuts which often contains wide families of strong inequalities forp 1 applications to the clique partitioning asymmetric traveling salesman plant location acyclic subgraph and linear ordering polytopes are briefly discussed

bc opt a branch and cut code for mixed integer programs a branch and cut mixed integer programming system called bc opt is described incorporating most of the valid inequalities that have been used or suggested for such systems namely lifted 0 1 knapsack inequalities 0 1 gub knapsack and integer knapsack inequalities flowcover and continuous knapsack inequalities path inequalities for fixed charge network flow structure and gomory mixed integer cuts the principal development is a set of interface routines allowing these cut routines to generate cuts for new subsets or aggregations of constraints

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

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

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

branch and cut for the maximum feasible subsystem problem this paper presents a branch and cut algorithm for the np hard maximum feasible subsystem problem for a given infeasible linear inequality system determine a feasible subsystem containing as many inequalities as possible the complementary problem where one has to remove as few inequalities as possible in order to make the system feasible can be formulated as a set covering problem the rows of this formulation correspond to irreducible infeasible subsystems which can be exponentially many it turns out that the main issue of a branch and cut algorithm for the maximum feasible subsystem problem max fs is to efficiently find such infeasible subsystems we present three heuristics for the corresponding np hard separation problem and discuss cutting planes from the literature such as set covering cuts of balas and ng gomory cuts and 0 frac 1 2 cuts furthermore we compare a heuristic of chinneck and a simple greedy algorithm the main contribution of this paper is an extensive computational study on a variety of instances arising in a number of applications

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

a decidability result about sufficient completeness of axiomatically specified abstract data types the problem of deciding whether an axiomatic specification of an abstract data type is sufficiently complete is known to be in general unsolvable regarding axioms as directed rewrite rules instead of symmetric equations a specification defines a reduction relation on terms it is proved that in the subclass of left linear axiomatic specifications the property of sufficient completeness is decidable if the corresponding reduction relation is normalizing and confluent the presented algorithm can also be used to determine a set of constructors for a specified data type

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

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

on splittable and unsplittable flow capacitated network design arc set polyhedra we study the polyhedra of splittable and unsplittable single arc set relaxations of multicommodity flow capacitated network design problems we investigate the optimization problems over these sets and the separation and lifting problems of valid inequalities for them in particular we give a linear time separation algorithm for the residual capacity inequalities 19 and show that the separation problem of c strong in equalities 7 is np hard but can be solved over the subspace of fractional variables only we introduce two classes of inequalities for the unsplittable flow problems we present a summary of computational experiments with a branch and cut algorithm for multicommodity flow capacitated network design problems to test the effectiveness of the results presented here empirically

sequence independent lifting for mixed integer programs with variable upper bounds we investigate the convex hull of the set defined by a single inequality with continuous and binary variables with variable upper bound constraints we extend the traditional flow cover inequality and show that it is valid for a restriction of the set in which some variables are fixed we also give conditions under which this inequality is facet defining and when it is not we show how it can be lifted to obtain valid inequalities for the entire set using sequence independent lifting in general computing the lifting function is np hard but under an additional restriction on the cover we obtain a closed form finally we show how these results imply and extend known results about the single node fixed charge flow polyhedron

valid inequalities based on simple mixed integer sets in this paper we use facets of simple mixed integer sets with three variables to derive a parametric family of valid inequalities for general mixed integer sets we call these inequalities two step mir inequalities as they can be derived by applying the simple mixed integer rounding mir principle of wolsey 1998 twice the two step mir inequalities define facets of the master cyclic group polyhedron of gomory 1969 in addition they dominate the strong fractional cuts of letchford and lodi 2002

optimal allocation of surgery blocks to operating rooms under uncertainty the allocation of surgeries to operating rooms ors is a challenging combinatorial optimization problem there is also significant uncertainty in the duration of surgical procedures which further complicates assignment decisions in this paper we present stochastic optimization models for the assignment of surgeries to ors on a given day of surgery the objective includes a fixed cost of opening ors and a variable cost of overtime relative to a fixed length of day we describe two types of models the first is a two stage stochastic linear program with binary decisions in the first stage and simple recourse in the second stage the second is its robust counterpart in which the objective is to minimize the maximum cost associated with an uncertainty set for surgery durations we describe the mathematical models bounds on the optimal solution and solution methodologies including an easy to implement heuristic numerical experiments based on real data from a large health care provider are used to contrast the results for the two models and illustrate the potential for impact in practice based on our numerical experimentation we find that a fast and easy to implement heuristic works fairly well on average across many instances we also find that the robust method performs approximately as well as the heuristic is much faster than solving the stochastic recourse model and has the benefit of limiting the worst case outcome of the recourse problem

sequence independent lifting for mixed knapsack problems with gub constraints in this paper we consider the semi continuous knapsack problem with generalized upper bound constraints on binary variables we prove that generalized flow cover inequalities are valid in this setting and under mild assumptions are facet defining inequalities for the entire problem we then focus on simultaneous lifting of pairs of variables the associated lifting problem naturally induces multidimensional lifting functions and we prove that a simple relaxation in a restricted domain is a superadditive function furthermore we also prove that this approximation is under extra requirements the optimal lifting function we then analyze the separation problem in two phases first finding a seed inequality and second select the inequality to be added in the first step we evaluate both exact and heuristic methods the second step is necessary because the proposed lifting procedure is simultaneous from where our class of lifted inequalities might contain an exponential number of these we choose a strategy of maximizing the resulting violation finally we test this class of inequalities using instances arising from electrical planning problems our tests show that the proposed class of inequalities is strong in the sense that the addition of these inequalities closes on average 57 70 of the root integrality gap and 97 70 of the relative gap while adding less than three cuts on average

valid inequalities based on the interpolation procedure we study the interpolation procedure of gomory and johnson 1972 which generates cutting planes for general integer programs from facets of cyclic group polyhedra this idea has recently been re considered by evans 2002 and gomory johnson and evans 2003 we compare inequalities generated by this procedure with mixed integer rounding mir based inequalities discussed in dash and gunluk 2003 we first analyze and extend the shooting experiment described in gomory johnson and evans we show that mir based inequalities dominate inequalities generated by the interpolation procedure in some important cases we also show that the gomory mixed integer cut is likely to dominate any inequality generated by the interpolation procedure in a certain probabilistic sense we also generalize a result of cornuejols li and vandenbussche 2003 on comparing the strength of the gomory mixed integer cut with related inequalities

manufacturer s mixed pallet design problem abstract we study a problem faced by a major beverage producer the company produces and distributes several brands to various customers from its regional distributors for some of these brands most customers do not have enough demand to justify full pallet shipments therefore the company decided to design a number of mixed or rainbow pallets so that its customers can order these unpopular brands without deviating too much from what they initially need we formally state the company s problem as determining the contents of a pre determined number of mixed pallets so as to minimize the total inventory holding and backlogging costs of its customers over a finite horizon we first show that the problem is np hard we then formulate the problem as a mixed integer linear program and incorporate valid inequalities to strengthen the formulation finally we use company data to conduct a computational study to investigate the efficiency of the formulation and the impact of mixed pallets on customers total costs

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

facets of two dimensional infinite group problems in this paper we lay the foundation for the study of the two dimensional mixed integer infinite group problem 2dmiigp we introduce tools to determine if a given continuous and piecewise linear function over the two dimensional infinite group is subadditive and to determine whether it defines a facet of 2dmiigp we then present two different constructions that yield the first known families of facet defining inequalities for 2dmiigp the first construction uses valid inequalities of the one dimensional integer infinite group problem 1diigp as building blocks for creating inequalities for the two dimensional integer infinite group problem 2diigp we prove that this construction yields all continuous piecewise linear facets of the two dimensional group problem that have exactly two gradients the second construction we present has three gradients and yields facet defining inequalities of 2dmiigp whose continuous coefficients are not dominated by those of facets of the one dimensional mixed integer infinite group problem 1dmiigp

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

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

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

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

safe bounds in linear and mixed integer linear programming current mixed integer linear programming solvers are based on linear programming routines that use floating point arithmetic occasionally this leads to wrong solutions even for problems where all coefficients and all solution components are small integers an example is given where many state of the art milp solvers fail it is then shown how using directed rounding and interval arithmetic cheap pre and postprocessing of the linear programs arising in a branch and cut framework can guarantee that no solution is lost at least for mixed integer programs in which all variables can be bounded rigorously by bounds of reasonable size

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

lifted inequalities for 0 1 mixed integer programming basic theory and algorithms we study the mixed 0 1 knapsack polytope which is defined by a single knapsack constraint that contains 0 1 and bounded continuous variables we develop a lifting theory for the continuous variables in particular we present a pseudo polynomial algorithm for the sequential lifting of the continuous variables and we discuss its practical use

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

on the foundations of parameter estimation for generalized partial linear models with b splines and continuous optimization generalized linear models are widely used in statistical techniques as an extension generalized partial linear models utilize semiparametric methods and augment the usual parametric terms with a single nonparametric component of a continuous covariate in this paper after a short introduction we present our model in the generalized additive context with a focus on the penalized maximum likelihood and the penalized iteratively reweighted least squares p irls problem based on b splines which is attractive for nonparametric components then we approach solving the p irls problem using continuous optimization techniques they have come to constitute an important complementary approach alternative to the penalty methods with flexibility for choosing the penalty parameter adaptively in particular we model and treat the constrained p irls problem by using the elegant framework of conic quadratic programming the method is illustrated using a small numerical example

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

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

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

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

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

on the strength of gomory mixed integer cuts as group cuts gomory mixed integer gmi cuts generated from optimal simplex tableaus are known to be useful in solving mixed integer programs further it is well known that gmi cuts can be derived from facets of gomory s master cyclic group polyhedron and its mixed integer extension studied by gomory and johnson in this paper we examine why cutting planes derived from other facets of master cyclic group polyhedra group cuts do not seem to be as useful when used in conjunction with gmi cuts for many practical problem instances we numerically show that once gmi cuts from different rows of the optimal simplex tableau are added to the formulation all other group cuts from the same tableau rows are satisfied

bend minimization in planar orthogonal drawings using integer programming we consider the problem of minimizing the number of bends in a planar orthogonal graph drawing while the problem can be solved via network flow for a given planar embedding of a graph it is np hard if we consider all planar embeddings our approach for biconnected graphs combines a new integer linear programming ilp formulation for the set of all embeddings of a planar graph with the network flow formulation of the bend minimization problem for fixed embeddings we report on extensive computational experiments with two benchmark sets containing a total of more than 12 000 graphs where we compared the performance of our ilp based algorithm with a heuristic and a previously published branch bound algorithm for solving the same problem our new algorithm is significantly faster than the previously published approach for the larger graphs of the benchmark graphs derived from industrial applications and almost twice as fast for the benchmark graphs from the artificially generated set of hard instances

n step mingling inequalities new facets for the mixed integer knapsack set the n step mixed integer rounding mir inequalities of kianfar and fathi math program 120 2 313 346 2009 are valid inequalities for the mixed integer knapsack set that are derived by using periodic n step mir functions and define facets for group problems the mingling and 2 step mingling inequalities of atamturk and gunluk math program 123 2 315 338 2010 are also derived based on mir and they incorporate upper bounds on the integer variables it has been shown that these inequalities are facet defining for the mixed integer knapsack set under certain conditions and generalize several well known valid inequalities in this paper we introduce new classes of valid inequalities for the mixed integer knapsack set with bounded integer variables which we call n step mingling inequalities for positive integer n these inequalities incorporate upper bounds on integer variables into n step mir and therefore unify the concepts of n step mir and mingling in a single class of inequalities furthermore we show that for each n the n step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions for n 2 we extend the results of atamturk and gunluk on facet defining properties of 2 step mingling inequalities for n 3 we present new facets for the mixed integer knapsack set as a special case we also derive conditions under which the n step mir inequalities define facets for the mixed integer knapsack set in addition we show that n step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets

proving weak properties of rewriting abstract in rule based programming properties of programs such as termination are in general considered in their strong acceptance i e on every computation branch but in practice they may hold in their weak acceptance only i e on at least one computation branch moreover weak properties are often enough to ensure that programs give the expected result there are very few results to handle weak properties of rewriting we address here two of them termination and reducibility to a constructor form in a unified framework allowing us to prove them inductively proof trees are developed which simulate rewriting trees by narrowing and abstracting subterms our technique is constructive in the sense that proof trees can be used to infer an evaluation strategy for any given input the right computation branch is developed without using a costly breadth first strategy nor backtracking

convex hull results for generalizations of the constant capacity single node flow set for single node flow sets with fixed costs and constant capacities on the inflow and outflow arcs a family of constant capacity flow covers are known to provide the convex hull in different special cases and are conjectured to provide it in the general case here we study more general mixed integer sets for which such single node flow cover inequalities suffice to give the convex hull in particular we consider the case of a path in which each node has one or several incoming and outgoing arcs with constant capacities and fixed costs this can be seen as a lot sizing set with production and sales decisions driven by costs and prices and by the lower and upper bounds on stocks instead of being driven by demands as in the standard lot sizing model the approach we take is classical we characterize the extreme points derive tight extended formulations and project out the additional variables specifically we show that fourier motzkin elimination though far from elegant can be used to carry out the non trivial projections the validity of the conjecture for the single node flow set follows from our results

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 the cardinality constrained knapsack problem a cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive this structure occurs for example in areas such as finance location and scheduling traditionally cardinality constraints are modeled by introducing auxiliary 0 1 variables and additional constraints that relate the continuous and the 0 1 variables we use an alternative approach in which we keep in the model only the continuous variables and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables to derive the valid inequalities we extend the concepts of cover and cover inequality commonly used in 0 1 programming to this class of problems and we show how cover inequalities can be lifted to derive facet defining inequalities we present three families of non trivial facet defining inequalities that are lifted cover inequalities finally we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0 1 variables over the traditional mip approach for this class of problems

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

generating facets for finite master cyclic group polyhedra using n step mixed integer rounding functions the n step mixed integer rounding mir functions generate n step mir inequalities for mip problems and are facets for the infinite group problems we show that the n step mir functions also directly generate facets for the finite master cyclic group polyhedra especially in many cases where the breakpoints of the n step mir function are not necessarily at the elements of the group hence the linear interpolation of the facet coefficients obtained has more than two slopes

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

generalized mixed integer rounding inequalities facets for infinite group polyhedra we present a generalization of the mixed integer rounding mir approach for generating valid inequalities for mixed integer programming mip problems for any positive integer n we develop n facets for a certain n 1 dimensional single constraint polyhedron in a sequential manner we then show that for any n the last of these facets which we call the n step mir facet can be used to generate a family of valid inequalities for the feasible set of a general mixed ip constraint which we refer to as the n step mir inequalities the gomory mixed integer cut and the 2 step mir inequality of dash and gunluk math program 105 1 29 53 2006 are the first two families corresponding to n 1 2 respectively the n step mir inequalities are easily produced using periodic functions which we refer to as the n step mir functions none of these functions dominates the other on its whole period finally we prove that the n step mir inequalities generate two slope facets for the infinite group polyhedra and hence are potentially strong

on the polyhedral structure of a multi item production planning model with setup times we present and study a mixed integer programming model that arises as a substructure in many industrial applications this model generalizes a number of structured mip models previously studied and it provides a relaxation of various capacitated production planning problems and other fixed charge network flow problems we analyze the polyhedral structure of the convex hull of this model as well as of a strengthened lp relaxation among other results we present valid inequalities that induce facets of the convex hull under certain conditions we also discuss how to strengthen these inequalities by using known results for lifting valid inequalities for 0 1 continuous knapsack problems

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

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

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

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

exponential lower bounds on the lengths of some classes of branch and cut proofs we examine the complexity of branch and cut proofs in the context of 0 1 integer programs we establish an exponential lower bound on the length of branch and cut proofs that use 0 1 branching and lift and project cuts called simple disjunctive cuts by some authors gomory chvatal cuts and cuts arising from the n0 matrix cut operator of lovasz and schrijver a consequence of the lower bound result in this paper is that branch and cut methods of the type described above have exponential running time in the worst case

cutting planes for integer programs with general integer variables we investigate the use of cutting planes for integer programs with general integer variables we show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0 1 variables we also explore the use of gomory s mixed integer cuts we address both theoretical and computational issues and show how to embed these cutting planes in a branch and bound framework we compare results obtained by using our cut generation routines in two existing systems with a commercially available branch and bound code on a range of test problems arising from practical applications 1998 the mathematical programming society inc published by elsevier science b v

the flow set with partial order the flow set with partial order is a mixed integer set described by a budget on total flow and a partial order on the arcs that may carry positive flow this set is a common substructure of resource allocation and scheduling problems with precedence constraints and robust network flow problems under demand capacity uncertainty we give a polyhedral analysis of the convex hull of the flow set with partial order unlike for the flow set without partial order cover type inequalities based on partial order structure are a function of a lifting sequence we study the lifting sequences and describe structural results on the lifting coefficients for general and simpler special cases we show that all lifting coefficients can be computed in polynomial time by solving maximum weight closure problems in general for the special case of induced minimal covers we give a sequence dependent characterization of the lifting coefficients we prove however if the partial order is defined by an arborescence then lifting is sequence independent and all lifting coefficients can be computed in linear time moreover if the partial order is defined by a path total order then the coefficients can be expressed explicitly we also give a complete polyhedral description of the flow set with partial order for the polynomially solvable total order case we show that finding an optimal lifting order for a given induced minimal cover and a given fractional solution is a submodular optimization problem which is solved greedily finally we present preliminary computational results with a cutting plane algorithm based on the lifting and separation results

outer product free sets for polynomial optimization and oracle based cuts this paper introduces cutting planes that involve minimal structural assumptions enabling the generation of strong polyhedral relaxations for a broad class of problems we consider valid inequalities for the set s cap p where s is a closed set and p is a polyhedron given an oracle that provides the distance from a point to s we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron these cuts are generated from convex forbidden zones or s free sets derived from the oracle we also consider the special case of polynomial optimization accordingly we develop a theory of outer product free sets where s is the set of real symmetric matrices of the form xx t all maximal outer product free sets of full dimension are shown to be convex cones and we identify several families of such sets these families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently computational experiments demonstrate the promise of our approach

fixed charge transportation problem facets of the projection polyhedron in this paper we consider the well known fixed charge transportation problem to send any flow from source si to destination tj we incur a unit variable shipping cost of cij and a fixed cost fij here we study the structure of the projection polyhedron of this problem in the space of 0 1 variables associated with fixed charges and we develop several classes of valid inequalities and derive conditions under which they are facet defining in some cases if the conditions are not satisfied we show how they can be lifted to define facets several heuristics for generating and adding these facets are presented using these results we develop a computationally effective algorithm for solving the problem the computational results clearly indicate the usefulness of this approach

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

the convex hull of two core capacitated network design problems the network loading problem nlp is a specialized capacitated network design problem in which prescribed point to point demand between various pairs of nodes of a network must be met by installing loading a capacitated facility we can load any number of units of the facility on each of the arcs at a specified arc dependent cost the problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost

mingling mixed integer rounding with bounds mixed integer rounding mir is a simple yet powerful procedure for generating valid inequalities for mixed integer programs when used as cutting planes mir inequalities are very effective for mixed integer programming problems with unbounded integer variables for problems with bounded integer variables however cutting planes based on lifting techniques appear to be more effective this is not surprising as lifting techniques make explicit use of the bounds on variables whereas the mir procedure does not in this paper we describe a simple procedure which we call mingling for incorporating variable bound information into mir by explicitly using the variable bounds the mingling procedure leads to strong inequalities for mixed integer sets with bounded variables we show that facets of mixed integer knapsack sets derived earlier by superadditive lifting techniques can be obtained by the mingling procedure in particular the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of marchand and wolsey math program 85 15 33 1999 as well as the continuous integer knapsack cover and pack inequalities of atamturk math program 98 145 175 2003 ann oper res 139 21 38 2005 in addition mingling inequalities give a generalization of the two step mir inequalities of dash and gunluk math program 105 29 53 2006 under some conditions

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

optimizing splitter and fiber location in a multilevel optical ftth network due to the emergence of bandwidth requiring services telecommunication operators are being compelled to renew their fix access network most of them favoring the fiber to the home ftth technology this paper focuses on the optimization of ftth deployment which is of prime importance due to the economic stakes the key design issue here is locating splitters and routing fi bers in an existing network infrastructure to which is associated a graph with given capacities on the edges no assumption is made on the structure of the graph first we propose a mixed integer formulation for this decision problem then valid inequalities and problem size reduction schemes are presented finally efficiency of solving approaches is assessed through extensive numerical tests performed on orange real life data

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

strong formulations for mixed integer programming a survey we attempt to motivate and survey recent research on the use of strong valid inequalities and reformulation to solve mixed integer programming problems

fixed charge transportation on a path optimization lp formulations and separation in the fixed charge transportation problem the goal is to optimally transport goods from depots to clients when there is a fixed cost associated to transportation or equivalently to opening an arc in the underlying bipartite graph we further motivate its study by showing that it is both a special case and a strong relaxation of the big bucket multi item lot sizing problem and a generalization of a simple variant of the single node flow set this paper is essentially a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path we give a mathcal o n 3 time optimization algorithm and a mathcal o n 2 size linear programming extended formulation we describe a new class of inequalities that we call path modular inequalities we give two distinct proofs of their validity the first one is direct and crucially relies on sub and super modularity of an associated set function the second proof is by showing that the projection of the extended linear programming formulations onto the original variable space yields exactly the polyhedron described by the path modular inequalities thus we also show that these inequalities suffice to describe the convex hull of the set of feasible solutions

lifting inequalities a framework for generating strong cuts for nonlinear programs in this paper we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs the theory is geometric and provides insights into lifting based cut generation procedures yielding short proofs of earlier results in mixed integer programming using convex extensions we obtain conditions that allow for sequence independent lifting in nonlinear settings paving a way for efficient cut generation procedures for nonlinear programs this sequence independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general purpose strong cuts for integer programs we specialize our lifting results to derive facet defining inequalities for mixed integer bilinear knapsack sets finally we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut generation techniques applied on a tight reformulation of the problem

a polyhedral study of integer variable upper bounds we study the polyhedron of the single node capacitated network design model with integer variable upper bounds we first give a characterization of valid inequalities that is useful in proving the validity of several classes of inequalities next we derive several classes of valid inequalities and we give conditions for them to be facet defining sequence independent lifting is used to obtain additional facets we conclude by reporting computational results with a branch and cut algorithm

tightening simple mixed integer sets with guaranteed bounds in this paper we study how to reformulate knapsack sets and simple mixed integer sets in order to obtain provably tight polynomially large formulations

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

tools for proving inductive equalities relative completeness and ω completeness abstract inductive theorems are properties valid in the initial algebra a now popular tool for proving them in equational theories or abstract data types is based on proof by consistency this method uses a completion procedure and requires two essential properties of the specification namely relative completeness and completeness this paper investigates ways of proving them for the first one the complement algorithm is presented it is based on unification and computation of coverings and complements for the second one a technique based on discrimination of pairs of normal forms is explained and illustrated through examples

conic mixed integer rounding cuts a conic integer program is an integer programming problem with conic constraints many problems in finance engineering statistical learning and probabilistic optimization are modeled using conic constraints here we study mixed integer sets defined by second order conic constraints we introduce general purpose cuts for conic mixed integer programming based on polyhedral conic substructures of second order conic sets these cuts can be readily incorporated in branch and bound algorithms that solve either second order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch and bound tree central to our approach is a reformulation of the second order conic constraints with polyhedral second order conic constraints in a higher dimensional space in this representation the cuts we develop are linear even though they are nonlinear in the original space of variables this feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed integer programming the reformulation also allows the use of polyhedral methods for conic integer programming we report computational results on solving unstructured second order conic mixed integer problems as well as mean variance capital budgeting problems and least squares estimation problems with binary inputs our computational experiments show that conic mixed integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed integer programs and hence improving their solvability

valid inequalities for problems with additive variable upper bounds we study the facial structure of a polyhedron associated with the single node relaxation of network flow problems with additive variable upper bounds this type of structure arises for example in production planning problems with setup times and in network certain expansion problems we derive several classes of valid inequalities for this polyhedron and give conditions under which they are facet defining our computational experience with large network expansion problems indicates that these inequalities are very effective in improving the quality of the linear programming relaxations