subset algebra lift operators for 0 1 integer programming we extend the sherali adams lovasz schrijver balas ceria cornuejols and lasserre lift and project methods for 0 1 optimization by considering liftings to subset algebras our methods yield polynomial time algorithms for solving a relaxation of a set covering problem at least as strong as that given by the set of all valid inequalities with small coefficients and more generally all valid inequalities where the right hand side is not very large relative to the positive coefficients in the left hand side applied to generalizations of vertex packing problems our methods yield in polynomial time relaxations that have unbounded rank using for example the n operator

aggregation based cutting planes for packing and covering integer programs in this paper we study the strength of chvatal gomory cg cuts and more generally aggregation cuts for packing and covering integer programs ips aggregation cuts are obtained as follows given an ip formulation we first generate a single implied inequality using aggregation of the original constraints then obtain the integer hull of the set defined by this single inequality with variable bounds and finally use the inequalities describing the integer hull as cutting planes our first main result is to show that for packing and covering ips the cg and aggregation closures can be 2 approximated by simply generating the respective closures for each of the original formulation constraints without using any aggregations on the other hand we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general ips the proof of the above stated results for the case of covering ips with bounds require the development of some new structural results which may be of independent interest finally we examine the strength of cuts based on k different aggregation inequalities simultaneously the so called multi row cuts and show that every packing or covering ip with a large integrality gap also has a large k aggregation closure rank in particular this rank is always at least of the order of the logarithm of the integrality gap

interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains abstract we investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions through formal derivations we show that nonlocal diffusion interface problems converge to their classical local counterparts in the limit of vanishing nonlocality a central focus of this paper is local nonlocal interface problems or interface problems with sharp transitions between local and nonlocal domains such problems can be cast as instances of a nonlocal interface problem with a finite horizon in certain regions and a vanishing horizon in other regions we derive a local nonlocal interface problem and utilize conservation principles to obtain local nonlocal interface conditions comparisons between nonlocal local and local nonlocal interface problems are presented analytically and numerically with a focus on multiscale aspects of nonlocal models induced by their inherent length scales

on the 0 1 facets of the set covering polytope in this paper we consider inequalities of the form jxj where j equals 0 or 1 and is a positive integer we give necessary and sufficient conditions for such inequalities to define facets of the set covering polytope associated with a 0 1 constraint matrixa these conditions are in terms of critical edges and critical cutsets defined in the bipartite incidence graph ofa and are in the spirit of the work of balas and zemel on the set packing problem where similar notions were defined in the intersection graph ofa furthermore we give a polynomial characterization of a class of 0 1 facets defined from chorded cycles of the bipartite incidence graph this characterization also yields all the 0 1 liftings of odd hole inequalities for the simple plant location polytope

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

enhancing an algorithm for set covering problems abstract in this note we present enhancements to a previously published algorithm for the optimal solution of set covering problems these enhancements relate to the use of a langrangean heuristic feasible solution exclusion constraints gomory f cuts and an improved branching strategy computational results for problems involving up to 400 rows and 4000 columns indicate that the enhanced algorithm gives better computational results than the original algorithm

on the rational polytopes with chvatal rank 1 abstract in this paper we study the following problem given a polytope p with chvatal rank 1 does p contain an integer point boyd and pulleyblank observed that this problem is in thecomplexity class np co np an indication that it is probably not np complete we solvethis problem in polynomial time for polytopes arising from the satisfiability problem of a givenformula with at least three literals per clause for simplices whose integer hull can be obtainedby adding at most a constant number of chvatal inequalities and for rounded polytopes weprove that any closed convex set whose chvatal closure is empty has an integer width of atmost n and we give an example showing that this bound is tight within an additive constantof 1 the promise that a polytope has chvatal rank 1 seems hard to verify though we provethat deciding emptiness of the chvatal closure of a given rational polytope p is np complete even when p is contained in the unit hypercube or is a rational simplex and even when p doesnot contain any integer point this has two implications i it is np hard to decide whethera given rational polytope p has chvatal rank 1 even when p is contained in the unit cube oris a rational simplex ii the optimization and separation problems over the chvatal closureof a given rational polytope contained in the unit hypercube or of a given rational simplex arenp hard these results improve earlier complexity results of cornuejols and li and eisenbrand finally we prove that for any positive integer k it is np hard to decide whether adding atmost k chvatal inequalities is sufficient to describe the integer hull of a given rational polytope

on the set covering polytope i all the facets with coefficients in 0 1 2 while the set packing polytope through its connection with vertex packing has lent itself to fruitful investigations little is known about the set covering polytope we characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0 1 or 2 and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining we show that all inequalities in the above class are contained in the elementary closure of the constraint set and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure we point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix finally we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation

critical cutsets of graphs and canonical facets of set packing polytopes a cutset in a graph g with node set n is the set of edges joining the nodes in a subset of n to those not in the subset we call a cutset critical if its removal produces a graph whose independence number is greater than that of g we use the concept of a critical cutset to give first a necessary then a sufficient condition for an inequality with 0 1 coefficients to be a facet of the set packing polytope defined by g

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

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

a model selection criterion for high dimensional linear regression statistical model selection is a great challenge when the number of accessible measurements is much smaller than the dimension of the parameter space we study the problem of model selection in the context of subset selection for high dimensional linear regressions accordingly we propose a new model selection criterion with the fisher information that leads to the selection of a parsimonious model from all the combinatorial models up to some maximum level of sparsity we analyze the performance of our criterion as the number of measurements grows to infinity as well as when the noise variance tends to zero in each case we prove that our proposed criterion gives the true model with a probability approaching one additionally we devise a computationally affordable algorithm to conduct model selection with the proposed criterion in practice interestingly as a side product our algorithm can provide the ideal regularization parameter for the lasso estimator such that lasso selects the true variables finally numerical simulations are included to support our theoretical findings

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

on the facial structure of the set covering polytope dimensional linear programming given a bipartite graphg v u e a cover ofg is a subset d subseteq v with the property that each nodeu u is adjacent to at least one nodev d if a positive weightcv is associated with each nodev v the covering problem cp is to find a cover ofg having minimum total weight

transitive packing a unifying concept in combinatorial optimization this paper attempts to provide a better understanding of the facial structure of polyhedra previously investigated separately it introduces the notion of transitive packing and the transitive packing polytope polytopes that turn out to be special cases of the transitive packing polytope include the node packing acyclic subdigraph bipartite subgraph planar subgraph clique partitioning partition transitive acyclic subdigraph interval order and relatively transitive subgraph polytopes we give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope thereby introducing generalized cycle generalized clique generalized antihole generalized antiweb and odd partition inequalities on the one hand these classes subsume several known classes of valid inequalities for several special cases on the other hand they yield many new inequalities for several other special cases for some of the classes we also prove a lower bound on their gomory chvatal rank finally we relate the concept of transitive packing to generalized set packing and covering as well as to balanced and ideal matrices

a branch and cut algorithm for the undirected traveling purchaser problem the purpose of this paper is to present a branch and cut algorithm for the undirectedtraveling purchaser problem which consists of determining a minimum cost route through a subset of markets where the cost is the sum of travel and purchase costs the problem is formulated as an integer linear program and several families of valid inequalities are derived to strengthen the linear relaxation the polyhedral structure of the formulation is analyzed and several classes of valid inequalities are proved to be facet defining a branch and cut procedure is developed and tested over four classes of randomly generated instances results show that the proposed algorithm outperforms all previous approaches and can optimally solve instances containing up to 200 markets

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

the bond based peridynamic system with dirichlet type volume constraint in this paper the bond based peridynamic system is analysed as a non local boundary value problem with volume constraint the study extends earlier works in the literature on non local diffusion and non local peridynamic models to include non positive definite kernels we prove the well posedness of both linear and nonlinear variational problems with volume constraints the analysis is based on some non local poincare type inequalities and the compactness of the associated non local operators it also offers careful characterizations of the associated solution spaces such as compact embedding separability and completeness in the limit of vanishing non locality the convergence of the peridynamic system to the classical navier equations of elasticity with poisson ratio is demonstrated

some facets of the simple plant location polytope we relate the simple plant location problem to the vertex packing problem and derive several classes of facets of their associated integer polytopes

on thermoelastostatics of composites with nonlocal properties of constituents ii estimation of effective material and field parameters abstract one considers a linear thermoelastic composite medium which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities it is assumed that the stress strain constitutive relations of constituents are described by the nonlocal integral operators whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity the general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained the method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as e g effective field hypothesis implicitly exploited in the known centering methods in a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties local or nonlocal of heterogeneities in such a case the levin s 1967 formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart a solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method the operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and ellipsoidal symmetry assumption quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented

lower bounds for lovasz schrijver systems and beyond follow from multiparty communication complexity we prove that an omega log 4 n lower bound for the three party number on the forehead nof communication complexity of the set disjointness function implies an n omega 1 size lower bound for treelike lovasz schrijver systems that refute unsatisfiable formulas in conjunctive normal form cnfs more generally we prove that an n omega 1 lower bound for the k 1 party nof communication complexity of set disjointness implies a 2 n omega 1 size lower bound for all treelike proof systems whose formulas are degree k polynomial inequalities

conditional model order estimation a new approach to model order selection is proposed based on the theory of sufficient statistics the method does not require any prior knowledge of the model parameters it is able to discriminate between models by basing the decision on the part of the data that is independent of the model parameters this is accomplished conceptually by transforming the data into a sufficient statistic and an ancillary statistic with respect to the model parameters it is the probability density function of the ancillary statistic when adjusted for its dimensionality that is used to estimate the order furthermore the rule is directly tied to the goal of minimizing the probability of error and does not employ any asymptotic approximations the estimator can be shown to be consistent and via computer simulation is found to outperform the minimum description length estimator

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

convergence studies in meshfree peridynamic simulations meshfree methods are commonly applied to discretize peridynamic models particularly in numerical simulations of engineering problems such methods discretize peridynamic bodies using a set of nodes with characteristic volume leading to particle based descriptions of systems in this paper we perform convergence studies of static peridynamic problems we show that commonly used meshfree methods in peridynamics suffer from accuracy and convergence issues due to a rough approximation of the contribution of nodes near the boundary of the neighborhood of a given node to numerical integrations we propose two methods to improve meshfree peridynamic simulations the first method uses accurate computations of volumes of intersections between neighbor cells and the neighborhood of a given node referred to as partial volumes the second method employs smooth influence functions with a finite support within peridynamic kernels numerical results demonstrate great improvements in accuracy and convergence of peridynamic numerical solutions when using the proposed methods

binary clutter inequalities for integer programs we introduce a new class of valid inequalities for general integer linear programs called binary clutter bc inequalities they include the 0 frac 1 2 cuts of caprara and fischetti as a special case and have some interesting connections to binary matroids binary clutters and gomory corner polyhedra we show that the separation problem for bc cuts is strongly hard in general but polynomially solvable in certain special cases as a by product we also obtain new conditions under which 0 frac 1 2 cuts can be separated in polynomial time these ideas are then illustrated using the traveling salesman problem tsp as an example this leads to an interesting link between the tsp and two apparently unrelated problems the t join and max cut problems

on some polytopes contained in the 0 1 hypercube that have a small chvatal rank in this paper we consider polytopes p that are contained in the unit hypercube we provide conditions on the set of 0 1 vectors not contained in p that guarantee that p has a small chvatal rank our conditions are in terms of the subgraph induced by these infeasible 0 1 vertices in the skeleton graph of the unit hypercube in particular we show that when this subgraph contains no 4 cycle the chvatal rank is at most 3 and when it has tree width 2 the chvatal rank is at most 4 we also give polyhedral decomposition theorems when this graph has a vertex cutset of size one or two

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

on thermoelastostatics of composites with nonlocal properties of constituents i general representations for effective material and field parameters abstract a theory of thermoelastic composites with nonlocal properties of constituents is analyzed for multiphase elastic solids of arbitrary geometry and material symmetry due to their generality one uses the nonlocal integral models because the gradient models are usually derived as approximations of corresponding integral models in the immediate infinitely closed vicinity of the point being considered one explores a simplified theory for linear macroscopically elasticity which differs from the classical local theory in the stress strain constitutive relation only whereas the equilibrium and compatibility equations remain unaltered one obtains the new representation of the effective modulus and compliance through the mechanical influence function which does not explicitly depend as opposed to its local counterpart on the elastic operators of constituents the representations for the effective eigenstrains and eigenstresses through either the mechanical influence functions or transformation influence functions are presented the effective strain energy and potential energy are expressed in terms of only average values of the state variables and the effective properties representations of both the first and second statistical moments of stress and strain fields in the constituents are also performed many of the results were obtained as the straightforward generalizations of their local counterparts because the methods used for obtaining the mentioned results widely exploit the hill s 1963 condition which holds for any compatible strain field and equilibrium stress field not necessarily related to each other by a specific stress strain relation

on the proper forms of bic for model order selection the bayesian information criterion bic is often presented in a form that is only valid in large samples and under a certain condition on the rate at which the fisher information matrix fim increases with the sample length this form has been improperly used previously in situations in which the conditions mentioned above do not hold in this correspondence we describe the proper forms of bic in several practically relevant cases that do not satisfy the above assumptions in particular we present a new form of bic for high signal to noise ratio snr cases the conclusion of this study is that bic remains one of the most successful existing rules for model order selection if properly used

generalized effective fields method in peridynamic micromechanics of random structure composites abstract we consider a static problem for statistically homogeneous matrix linear bond based peridynamic composite materials cms for the media subjected to remote homogeneous volumetric boundary loading one proved that the effective behavior of this media is governed by conventional effective constitutive equation as in the local elasticity theory it was made by the most exploitation of the popular tools and concepts used in conventional elasticity of composite materials cms with their adaptation to peridynamics this is extraction from the material properties a constituent of the matrix properties effective moduli are expressed through the introduced new notion of the average local polarization tensor the average is performed over the surface of the extended inclusion phase rather than over an entire space any spatial derivatives of displacement fields are not required the basic hypotheses of locally elastic micromechanics are generalized to their peridynamic counterparts in particular in the generalized effective field method efm proposed the effective field is evaluated from self consistent estimations by the use of closing of a corresponding integral equation in the framework of the quasi crystalline approximation in so doing the classical effective field hypothesis is relaxed and the hypothesis of the ellipsoidal symmetry of the random structure of cms is not used one demonstrates some similarity and difference with respect to other methods the dilute approximation and mori tanaka approach proposed before in micromechanics of peridynamic cms comparative numerical analyses of these methods are performed for 1d case

optimal solution of set covering partitioning problems using dual heuristics we present an algorithm for a mixed set covering partitioning model that includes as special cases the well known set covering problem and set partitioning problem the novel feature of our algorithm is the use of continuous heuristics applied to the dual of the linear programming relaxation to provide lower bounds for a branch and bound algorithm the heuristics are continuous adaptations of the well known greedy and 3 opt methods that have been applied to a variety of combinatorial optimization problems our algorithm has outperformed the current best set covering algorithm of balas and ho 1980 by about a factor of 3 and appears to improve on the best existing set partitioning algorithm by more than an order of magnitude

asymptotic map criteria for model selection the two most popular model selection rules in signal processing literature have been akaike s 1974 criterion aic and rissanen s 1978 principle of minimum description length mdl these rules are similar in form in that they both consist of data and penalty terms their data terms are identical but the penalties are different mdl being more stringent toward overparameterization aic penalizes for each additional model parameter with an equal incremental amount of penalty regardless of the parameter s role in the model in most of the literature on model selection mdl appears in a form that also suggests equal penalty for every unknown parameter this mdl criterion we refer to as naive mdl in this paper we show that identical penalization for every parameter is not appropriate and that the penalty has to depend on the model structure and type of model parameters the approach to showing this is bayesian and it relies on large sample theory we derive maximum a posteriori map rules for several different families of competing models and obtain forms that are similar to aic and naive mdl for some families however we find that the derived penalties are different in those cases our extensive simulations show that the map rule outperforms aic and naive mdl

on the maximum feasible subsystem problem iiss and iis hypergraphs we consider the max fs problem for a given infeasible linear system ax b determine a feasible subsystem containing as many inequalities as possible this problem which is np hard and also difficult to approximate has a number of interesting applications in a wide range of fields in this paper we examine structural and algorithmic properties of max fs and of irreducible infeasible subsystems iiss which are intrinsically related since one must delete at least one constraint from each iis to attain feasibility first we provide a new simplex decomposition characterization of iiss and prove that finding a smallest cardinality iis is very difficult to approximate then we discuss structural properties of iis hypergraphs i e hypergraphs in which each edge corresponds to an iis and show that recognizing iis hypergraphs subsumes the steinitz problem for polytopes and hence is np hard finally we investigate rank facets of the feasible subsystem polytope whose vertices are incidence vectors of feasible subsystems of a given infeasible system in particular using the iis hypergraph structural result we show that only two very specific types of rank inequalities induced by generalized antiwebs which generalize cliques odd holes and antiholes to general independence systems can arise as facets

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

convergence adaptive refinement and scaling in 1d peridynamics we introduce here adaptive refinement algorithms for the non local method peridynamics which was proposed in j mech phys solids 2000 48 175 209 as a reformulation of classical elasticity for discontinuities and long range forces we use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected we discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1d in the limit of the horizon going to zero continuous micromoduli lead to optimal rates of convergence independent of the grid used while discontinuous micromoduli produce optimal rates of convergence only for uniform grids examples for static and dynamic elasticity problems in 1d are shown the relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement for the same number of nodes copyright 2008 john wiley sons ltd

the steiner connectivity problem the steiner connectivity problem has the same significance for line planning in public transport as the steiner tree problem for telecommunication network design it consists in finding a minimum cost set of elementary paths to connect a subset of nodes in an undirected graph and is therefore a generalization of the steiner tree problem we propose an extended directed cut formulation for the problem which is in comparison to the canonical undirected cut formulation provably strong implying e g a class of facet defining steiner partition inequalities since a direct application of this formulation is computationally intractable for large instances we develop a partial projection method to produce a strong relaxation in the space of canonical variables that approximates the extended formulation we also investigate the separation of steiner partition inequalities and give computational evidence that these inequalities essentially close the gap between undirected and extended directed cut formulation using these techniques large steiner connectivity problems with up to 900 nodes can be solved within reasonable optimality gaps of typically less than five percent

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

fea in elasticity of random structure composites reinforced by heterogeneities of non canonical shape abstract one considers linearly elastic composite media which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non canonical shape effective elastic moduli as well as the first statistical moments of stresses in the phases are estimated the explicit new representations of the effective moduli and stress concentration factors are expressed through some building block described by numerical solution for one heterogeneity inside the infinite medium subjected to homogeneous remote loading the method uses as a background a new general integral equation proposed in buryachenko 2010a buryachenko 2010b which incorporates influence of stress inhomogeneity inside the inclusion on the effective field and makes it possible to reconsider basic concepts of micromechanics such as effective field hypothesis quasi crystalline approximation and the hypothesis of ellipsoidal symmetry the results of this reconsideration are quantitatively estimated for some modeled composite reinforced by aligned homogeneous heterogeneities of non canonical shape some new effects are detected that are impossible in the framework of a classical background of micromechanics

0 1 polytopes with quadratic chvatal rank for a polytope p the chvatal closure p p is obtained by simultaneously strengthening all feasible inequalities cx with integral c to cx the number of iterations of this procedure that are needed until the integral hull of p is reached is called the chvatal rank if p 0 1 n then it is known that o n2 log n iterations always suffice and at least 1 1 e o 1 n iterations are sometimes needed leaving a huge gap between lower and upper bounds we prove that there is a polytope contained in the 0 1 cube that has chvatal rank n2 closing the gap up to a logarithmic factor in fact even a superlinear lower bound was mentioned as an open problem by several authors our choice of p is the convex hull of a semi random knapsack polytope and a single fractional vertex the main technical ingredient is linking the chvatal rank to simultaneous diophantine approximations w r t the 1 norm of the normal vector defining p

model order selection rules for covariance structure classification in radar the adaptive classification of the interference covariance matrix structure for radar signal processing applications is addressed in this paper this represents a key issue because many detection architectures are synthesized assuming a specific covariance structure which may not necessarily coincide with the actual one due to the joint action of the system and environment uncertainties the considered classification problem is cast in terms of a multiple hypotheses test with some nested alternatives and the theory of model order selection mos is exploited to devise suitable decision rules several mos techniques such as the akaike takeuchi and bayesian information criteria are adopted and the corresponding merits and drawbacks are discussed at the analysis stage illustrating examples for the probability of correct model selection are presented showing the effectiveness of the proposed rules

regenerator location problem polyhedral study and effective branch and cut algorithms abstract in this paper we study the regenerator location problem rlp this problem arises in optical networks where an optical signal can only travel a certain maximum distance called the optical reach before its quality deteriorates needing regenerations by regenerators deployed at network nodes the rlp is to determine a minimal number of network nodes for regenerator placement such that for each node pair there exists a path of which no sub path without internal regenerators has a length greater than the optical reach starting with a set covering formulation involving an exponential number of constraints reported and studied in rahman 2012 and aneja 2012 we study the facial structure of the polytope arising from this formulation significantly extending known results making use of these polyhedral results we present a new branch and cut b c solution approach to solve the rlp to optimality we present a series of computational experiments to evaluate two versions of the proposed b c approach over 400 benchmark rlp instances we first compare them with an available exact method for the rlp in the literature because of the equivalence among the rlp the minimum connected dominating set problem mcdsp and the maximum leaf spanning tree problem mlstp we further compare our approaches with other available exact algorithms using 47 benchmark mcdsp mlstp instances

lifted cycle inequalities for the asymmetric traveling salesman problem we investigate the family of facet defining inequalities for the asymmetric traveling salesman ats polytope obtainable by lifting the cycle inequalities we establish several properties of this family that earmark it as arguably the most important among the asymmetric inequalities for the ats polytope known to date in particular we give several results that characterize the pattern of lifting coefficients and thereby facilitate the identification of facet defining inequalities this characterization is used to investigate three new large classes of lifted cycle inequalities that we call shell fork and curtain inequalities these latter inequalities have unbounded chvatal rank furthermore the pattern of lifting coefficients that we identify makes it easy to develop efficient separation routines finally each member of the family is shown to have a counterpart for the symmetric ts sts polytope that is often new and is obtainable by mapping the inequality for the ats polytope into a certain face of the sts polytope and then lifting the resulting inequality into one for the sts polytope itself

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

transience bounds for long walks let g v e be a directed graph with rewards re for e e e we show that if there exists an s t walk in g of length k v 2 then there exists a maximum reward s t walk of length k with at least k v 2 arcs from a cycle with maximal mean reward if a is the reward incidence matrix of g then the maximum rewards of s t walks of length k can be determined by computing the k fold matrix product a k where the operations plus and times are replaced by max and plus denoted by and max algebra dynamic systems y k y k 1 a have been used to model certain transportation and automated manufacturing systems and can be interpreted as deterministic dynamic programming recursions if g is strongly connected then y k da y k da 1 for all k ka where is the maximum cycle mean in g and ka and da are the max algebra transient and period cyclicity of a for given y 0 the transient and period of the system may satisfy ka and da we derive a tight bound on ka and give polynomial algorithms for computing ka and and for solving deterministic dynamic programming problems we also give analogous results for walks of maximum discounted reward

solution of general integral equations of micromechanics of heterogeneous materials one considers a linear composite materials cm which consists of a homogeneous matrix containing a random set of heterogeneities an operator form of solution of the general integral equation gie for the general cases of local and nonlocal problems static and wave motion phenomena for composite materials with random statistically homogeneous and inhomogeneous so called graded structures containing coated or uncoated inclusions of any shape and orientation with perfect and imperfect interfaces and subjected to any number of coupled or uncoupled homogeneous or inhomogeneous external fields of different physical nature the gie connecting the driving fields and fluxes in a point being considered and the fields in the surrounding points are obtained for the random fields of heterogeneities in the infinite media estimations of the effective properties and both the first and second statistical moments of fields in the constituents of cms are presented in a general form of perturbations introduced by the heterogeneities and taking into account a possible imperfection of interface conditions the solution methods of gies are obtained without any auxiliary assumptions such as the effective field hypothesis efh which is implicitly exploited in the known methods of micromechanics some particular cases asymptotic representations and simplifications of proposed methods are presented for the particular constitutive equations such as linear thermoelastic cases with the perfect and imperfect interfaces conductivity problem problems for piezoelectric and other coupled phenomena composites with nonlocal elastic properties of constituents and the wave propagation in composites with electromagnetic optic and mechanical responses

on the matrix cut rank of polyhedra lovasz and schrijver 1991 described a semidefinite operator for generating strong valid inequalities for the 0 1 vectors in a prescribed polyhedron among their results they showed that n iterations of the operator are sufficient to generate the convex hull of 0 1 vectors contained in a polyhedron in n space we give a simple example having chvatal rank 1 that meets this worst case bound of n we describe another example requiring n iterations even when combining the semidefinite and gomory chvatal operators this second example is used to show that the standard linear programming relaxation of a k city traveling salesman problem requires at least k 8 iterations of the combined operator this bound is best possible up to a constant factor as k 1 iterations suffice

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

an exact algorithm and a metaheuristic for the multi vehicle covering tour problem with a constraint on the number of vertices abstract the multi vehicle covering tour problem m ctp involves finding a minimum length set of vehicle routes passing through a subset of vertices subject to constraints on the length of each route and the number of vertices that it contains such that each vertex not included in any route lies within a given distance of a route this paper tackles a particular case of m ctp where only the restriction on the number of vertices is considered i e the constraint on the length is relaxed the problem is solved by a branch and cut algorithm and a metaheuristic to develop the branch and cut algorithm we use a new integer programming formulation based on a two commodity flow model the metaheuristic is based on the evolutionary local search els method proposed in 23 computational results are reported for a set of test problems derived from the tsplib

computation of the lasserre ranks of some polytopes over the years various lift and project methods have been proposed to construct hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p rn that converge to p in n steps many such methods have been shown to require n steps in the worst case in this paper we show that the method of lasserre also requires n steps in the worst case

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

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

optimizing over the subtour polytope of the travelling salesman problem a commonly studied relaxation of the travelling salesman problem is obtained by adding subtour elimination constraints to the constraints of a 2 factor problem and removing the integrality requirement we investigate the problem of solving this relaxation for a special type of objective function we also discuss some ways in which this relates to the concept of rank introduced by chvatal

on the acyclic subgraph polytope the acyclic subgraph problem can be formulated as follows given a digraph with arc weights find a set of arcs containing no directed cycle and having maximum total weight we investigate this problem from a polyhedral point of view and determine several classes of facets for the associated acyclic subgraph polytope we also show that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time this implies that the acyclic subgraph problem can be solved in polynomial time for weakly acyclic digraphs this generalizes a result of lucchesi for planar digraphs

testing the odd bicycle wheel inequalities for the bipartite subgraph polytope in their paper on facets of the bipartite subgraph polytope barahona grotschel and mahjoub pose the question is there a polynomial time algorithm testing the odd bicycle wheel inequalities this note describes such an algorithm

rank inequalities and separation algorithms for packing designs and sparse triple systems combinatorial designs find numerous applications in computer science and are closely related to problems in coding theory packing designs correspond to codes with constant weight 4 sparse partial steiner triple systems 4 sparse pstss correspond to erasure resilient codes that are useful in handling failures in large disk arrays chee colbourn ling discrete appl math to appear hellerstein gibson karp katz paterson algorithmica 12 1994 182 208 the study of polytopes associated with combinatorial problems has proven to be important for both algorithms and theory but only recently the study of design polytopes has been pursued moura math appl 368 1996 227 254 moura ph d thesis university of toronto 1999 moura proc seventh annu european symp prague 1999 lecture notes in computer science vol 1643 springer berlin 1999 pp 462 475 wengrzik master s thesis universitat berlin 1995 zehendner doctoral thesis universitat augsburg 1986 in this article we study polytopes associated with t v k packing designs and with m sparse pstss subpacking and l sparseness inequalities are introduced and studied they can be regarded as rank inequalities for the independence systems associated with these designs conditions under which subpacking inequalities define facets are derived in particular those which define facets for pstss are determined for m 4 the l sparseness inequalities with 2 l m are proven to induce facets for the m sparse psts polytope this proof uses extremal families of pstss known as erdos configurations separation algorithms for these inequalities are proposed we incorporate some of the sparseness inequalities in a polyhedral algorithm and determine maximal 4 sparse psts v v 16 an upper bound on the size of m sparse pstss is presented

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

facets and lifting procedures for the set covering polytope given a family of subsets of an arbitrary groundsete acover of is any setc e having non empty intersection with every subset in in this paper we deal with thecovering polytope i e the convex hull of the incidence vectors of all the covers of in section 2 we review all the known properties of the covering polytope in sections 3 and 4 we introduce two new classes of non boolean facets of such a polytope in sections 5 and 6 we describe some non sequential lifting procedures in section 7 a generalization of the notion ofweb introduced by l e trotter is presented together with the facets of the covering polytope produced by such a structure moreover the strong connections between several combinatorial problems and the covering problem are pointed out and exploiting those connections some examples are presented of new facets for the knapsack and acyclic subdigraph polytopes

green s functions in non local three dimensional linear elasticity in this paper we compare small deformations in an infinite linear elastic body due to the presence of point loads within the classical local formulation to the corresponding deformations in the peridynamic non local formulation owing to the linearity of the problem the response to a point load can be used to obtain the response to general body force loading functions by superposition using laplace and fourier transforms we thus obtain an integral representation for the three dimensional peridynamic solution with the help of green s functions we illustrate this new theoretical result by dynamic and static examples in one and three dimensions in addition to this main result we also derive the non local three dimensional jump conditions as well as the weak formulation of peridynamics together with the associated finite element discretization

high snr consistent linear model order selection and subset selection high snr consistency of model order selection criteria in linear regression models has attracted a lot of attention in the signal processing community recently it is now known that exponentially embedded family versions of minimum description length etc are high snr consistent however a general framework for high snr consistency in linear regression is still missing this paper fills this gap by deriving necessary and sufficient conditions nscs for model order selection criteria in both known and unknown noise variance situations to be high snr consistent many popular model order selection criteria are proved to be high snr consistent using these nscs we also provide a convergence rate analysis to discriminate between various high snr consistent model order selection criteria a direct application of model order selection techniques to the very important subset selection problem is computationally infeasible this paper establish the high snr consistency of an existing t statistics based index ordering scheme that allows the conversion of a subset selection problem into a model order selection problem combining this index ordering with high snr consistent model order selection criteria leads to a novel subset selection procedure ssp that is numerically shown to outperform existing high snr consistent ssps

inconsistency of the mdl on the performance of model order selection criteria with increasing signal to noise ratio in the problem of model order selection it is well known that the widely used minimum description length mdl criterion is consistent as the sample size n however the consistency as the noise variance 2 0 has not been studied in this paper we find that the mdl is inconsistent as 2 0 the result shows that the mdl has a tendency to overestimate the model order we also prove that another criterion the exponentially embedded family eef is consistent as 2 0 therefore in a high signal to noise snr scenario the eef provides a better criterion to use for model order selection

new formulations for the uncapacitated multiple allocation hub location problem in this paper we review the integer linear formulations of the uncapacitated multiple allocation hub location problem we study the scope of validity of these formulations and give new ones that generalize the older formulations our formulations allow one or two visits to hubs and include a more general cost structure that needs not satisfy the triangle inequality we prove that the constraints defined by cliques of a related intersection graph are tighter constraints than the classical ones we also discuss a pre processing of the problem which is very useful for reducing its size finally we check the strength of the new formulations and compare them with others in the literature by solving instances of two commonly used data sets the cab civil aeronautics board and ap australian post and also randomly generated instances our computational results clearly show that our formulations outperform all others previously used for small and medium problems

a generalization of antiwebs to independence systems and their canonical facets we consider independence system polytopes i e polytopes whose extreme points are the incidence vectors of the sets of an independence system we first give a sufficient condition for recognizing boolean facets then the notion of antiweb introduced by trotter for graphs is generalized to independence systems and used for obtaining canonical facets of the associated polytopes we also point out how our results relate with known ones for knapsack set covering and matroid polytopes

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

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

on the set covering polytope ii lifting the facets with coefficients in 0 1 2 in an earlier paper mathematical programming 43 1989 57 69 we characterized the class of facets of the set covering polytope defined by inequalities with coefficients equal to 0 1 or 2 in this paper we connect that characterization to the theory of facet lifting in particular we introduce a family of lower dimensional polytopes and associated inequalities having only three nonzero coefficients whose lifting yields all the valid inequalities in the above class with the lifting coefficients given by closed form expressions