very large scale linear programming a case study in combining interior point and simplex methods experience with solving a 12 753 313 variable linear program is described this problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application a scheme is described that requires successive solutions of small subproblems yielding a procedure that has little growth in solution time in terms of the number of variables experience using the simplex method as implemented in cplex an interior point method as implemented in obi and a hybrid interior point simplex approach is reported the resulting procedure illustrates the power of an interior point simplex combination for solving very large scale linear programs

the realization of hyperelliptic curves through endomorphisms of kronecker modules abstract let k be an algebraically closed field and a the kronecker algebra over k a general problem is to study the endomorphism algebras of a modules m that are extensions of finite dimensional torsion free rank one a modules p by infinite dimensional torsion free rank one a modules n such endomorphism algebras can be studied by means of a quadratic polynomial f y in one variable y over the rational function field k x we call this f y the regulator of the extension we prove that if the regulator has non zero discriminant then end m is a noetherian commutative k algebra we also prove that subject to a regulator with non zero discriminant end m is affine over k if and only if end n is affine in which case end m is the coordinate ring of a hyperelliptic curve

computing redundant resources for the resource constrained project scheduling problem several efficient lower bounds and time bound adjustment methods for the resource constrained project scheduling problem rcpsp have recently been proposed some of them are based on redundant resources in this paper we define redundant functions which are very useful for computing redundant resources we also describe an algorithm for computing all maximal redundant functions once all these redundant functions have been determined we have to identify those that are useful for bounding surprisingly their number is reasonable even for large resource capacities so a representative subset of them can be tabulated to be used efficiently computational results on classical rcpsp instances confirm their usefulness

a branch and price and cut approach for sustainable crop rotation planning in this paper we study a multi periodic production planning problem in agriculture this problem belongs to the class of crop rotation planning problems which have received increased attention in the literature in recent years crop cultivation and fallow periods must be scheduled on land plots over a given time horizon so as to minimize the total surface area of land used while satisfying crop demands every period this problem is proven strongly np hard we propose a 0 1 linear programming compact formulation based on crop sequence graphs an extended formulation is then provided with a polynomial time pricing problem and a branch and priceand cut bpc algorithm is presented with adapted branching rules and cutting planes the numerical experiments on instances varying the number of crops periods and plots show the effectiveness of the bpc for the extended formulation compared to solving the compact formulation even though these two formulations have the same linear relaxation bound

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

a new lp based lower bound for the cumulative scheduling problem abstract this paper deals with the computation of lower bounds for cumulative scheduling problems based on a new linear programming formulation these lower bounds take into account how resource requirements can be satisfied simultaneously for a given resource capacity one of the main interests of this paper is that the solutions of the lp can be tabulated for a given value of resource capacity thus even if it is based on a linear programming formulation the computation of the bounds is low time consuming as confirmed by our computational results on the resource constrained project scheduling problem

the cutting stock problem and integer rounding an integer programming problem is said to have the integer round up property if its optimal value is given by the least integer greater than or equal to the optimal value of its linear programming relaxation in this paper we prove that certain classes of cutting stock problems have the integer round up property the proof of these results relies upon the decomposition properties of certain knapsack polyhedra

an exact algorithm for the resource constrained project scheduling problem based on a new mathematical formulation in this paper we consider the project scheduling problem with resource constraints where the objective is to minimize the project makespan we present a new 0 1 linear programming formulation of the problem that requires an exponential number of variables corresponding to all feasible subsets of activities that can be simultaneously executed without violating resource or precedence constraints different relaxations of the above formulation are used to derive new lower bounds which dominate the value of the longest path on the precedence graph and are tighter than the bound proposed by stinson et al 1978 a tree search algorithm based on the above formulation that uses new lower bounds and dominance criteria is also presented computational results indicate that the exact algorithm can solve hard instances that cannot be solved by the best algorithms reported in the literature

deflation projector preconditioning and balancing in iterative substructuring methods connections and new results in this paper projector preconditioning also known as the deflation method as well as the balancing preconditioner are applied to the dual primal finite element tearing and interconnecting feti dp and balancing domain decomposition by constraints bddc methods in order to create a second independent coarse problem this may help to extend the parallel scalability of classical feti dp and bddc methods without the use of inexact solvers and may also be used to improve the robustness e g for almost incompressible elasticity problems connections of feti dp methods applying a transformation of basis using a larger coarse space with a corresponding feti dp method using projector preconditioning or balancing are pointed out it is then shown that the methods have essentially the same spectrum numerical results for compressible and almost incompressible linear elasticity are provided the sensitivity of the projection methods to an inexact computation of the projections is numerically investigated and a different behavior for projector preconditioning and the balancing preconditioner is found

an adaptive bddc method enhanced with prior selected primal constraints abstract an adaptive bddc balancing domain decomposition by constraints method is considered for three dimensional elliptic problems with coefficients of random variation and high contrast for such model problems a certain generalized eigenvalue problem on each subdomain interface is formed to select primal constraints adaptively in three dimensions eigenvalue problems are formed on each face or edge nodal equivalence classes for the case of edges proposed eigenvalue problems in previous studies are not satisfactory while those for faces perform very effectively the eigenvalue problems can be enhanced by utilizing prior selected primal constraints in this paper this new idea is adopted when forming edge eigenvalue problems and the resulting adaptive bddc preconditioner is analyzed in numerical experiments the new edge eigenvalue problem is also shown to provide a more optimal set of adaptive primal constraints compared to those in the previous studies

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

feti dp methods with an adaptive coarse space a coarse space is constructed for the dual primal finite element tearing and interconnecting feti dp domain decomposition method applied to highly heterogeneous problems by solving local generalized eigenvalue problems for certain problems with highly varying coefficients e g from multiscale simulations the coefficient jump will appear in the condition number bound even if standard techniques such as scaling and the weighting of constraints are used the feti dp theory is revisited and two central estimates are identified where the dependency on the coefficient contrast can enter the condition number bound the first is a poincare inequality and the second an extension theorem these estimates are replaced by local eigenvalue problems enriching the feti dp coarse space by a few numerically computed eigenvectors yields independence of the contrast of the coefficients even in challenging situations

a branch and bound algorithm for the resource constrained project scheduling problem a branch and bound algorithm is presented for the resource constrained project scheduling problem rcpsp given are n activities which have to be processed without preemptions during the processing period of an activity constant amounts of renewable resources are needed where the available capacity of each resource type is limited furthermore finish start precedence relations between the activities are given the objective is to determine a schedule with minimal makespan the branching scheme starts from a graph representing a set of conjunctions the classical finish start precedence constraints and disjunctions induced by the resource constraints then it either introduces disjunctive constraints between pairs of activities or places these activities in parallel concepts of immediate selection are developed in connection with this branching scheme immediate selection allows to add conjunctions as well as further disjunctions and parallelity relations computational results based on the test data of kolisch et al kolisch r sprecher a drexl a 1995 management science 41 1693 1703 and kolisch and sprecher kolisch r sprecher a 1997 psplib a project scheduling problem library ejor 96 pp 205 216 are reported

solving the resource constrained project scheduling problem using filter and fan approach the resource constrained project scheduling problem is a notoriously difficult problem in combinatorial optimization for which extensive investigation has been devoted to the development of efficient algorithms to find optimal or near optimal solutions this paper proposes a new heuristic algorithm for this problem based on filter and fan method incorporated with a local search exploring in the defined neighborhood space in the algorithm the local search is used to generate a starting solution as well as to re optimize the best schedules produced by the filter and fan method the filter and fan is itself a local search procedure that generates compound moves in a tree search fashion computational results applied on a standard set of 2040 benchmark problems from the literature demonstrate the effectiveness of the approach

a parallel implementation of dual primal feti methods for three dimensional linear elasticity using a transformation of basis dual primal feti methods for linear elasticity problems in three dimensions are considered these are nonoverlapping domain decomposition methods where some primal continuity constraints across subdomain boundaries are required to hold throughout the iterations whereas most of the constraints are enforced by lagrange multipliers an algorithmic framework for dual primal feti methods is described together with a transformation of basis to implement the primal constraints numerical results obtained from a parallel implementation of these algorithms applied to a model benchmark problem with structured meshes and to problems with more complicated geometries from industrial and biological applications using unstructured meshes are provided these results show that the presented dual primal feti algorithms are numerical and parallel scalable

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

the classification problem for torsion free abelian groups of finite rank in 1937 baer 5 introduced the notion of the type of an element in a torsion free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion free abelian groups of rank 1 since then despite the efforts of such mathematicians as kurosh 23 and malcev 25 no satisfactory system of complete invariants has been found for the torsion free abelian groups of finite rank n 2 so it is natural to ask whether the classification problem is genuinely more difficult for the groups of rank n 2 of course if we wish to show that the classification problem for the groups of rank n 2 is intractible it is not enough merely to prove that there are 2 such groups up to isomorphism for there are 2 pairwise nonisomorphic groups of rank 1 and we have already pointed out that baer has given a satisfactory classification for this class of groups in this paper following friedman stanley 11 and hjorth kechris 15 we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n 2 recall that up to isomorphism the torsion free abelian groups of rank n are exactly the additive subgroups of the n dimensional vector space qf which contain n linearly independent elements thus the collection of torsion free abelian groups of rank 1 r n can be naturally identified with the set s qn of all nontrivial additive subgroups of qn notice that s qn is a borel subset of the polish space p qn of all subsets of qn and hence s qn can be regarded as a standard borel space i e a polish space equipped with its associated o algebra of borel subsets here we are identifying p qn with the space 2 qn of all functions h qn 0 1 equipped with the product topology furthermore the natural action of gln q on the vector space qn induces a corresponding borel action on s qn and it is easily checked that if a b c s qn then a v b iff there exists an element f e gln q such that p a b it follows that the isomorphism relation on s qn is a countable borel equivalence relation if x is a standard borel space then a borel equivalence relation on x is an equivalence relation e c x2 which is a borel subset of x2 the borel equivalence relation e is said to be countable iff every e equivalence class is countable

isogeometric bddc preconditioners with deluxe scaling a balancing domain decomposition by constraints bddc preconditioner with a novel scaling introduced by dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling is extended to isogeometric analysis of scalar elliptic problems this new scaling turns out to be more powerful than the standard rho and stiffness scalings considered in a previous isogeometric bddc study our h analysis shows that the condition number of the resulting deluxe bddc preconditioner is scalable with a quasi optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings in particular for large isogeometric polynomial degree and high regularity

adaptive coarse spaces for feti dp in three dimensions an adaptive coarse space approach including a condition number bound for dual primal finite element tearing and interconnecting feti dp methods applied to three dimensional problems with coefficient jumps inside subdomains and across subdomain boundaries is presented the approach is based on a known adaptive coarse space approach enriched by a small number of additional local edge eigenvalue problems these edge eigenvalue problems serve to make the method robust and permit a condition number bound which depends only on the tolerance of the local eigenvalue problems and some properties of the domain decomposition the introduction of the edge eigenvalue problems thus turns a well known condition number indicator for feti dp and balancing domain decomposition by constraints bddc methods into a condition number estimate numerical results are presented for linear elasticity and heterogeneous materials supporting our theoretical findings the problems considered include those with random coefficients an

a linear programming and constraint propagation based lower bound for the rcpsp abstract a new destructive lower bound for the resource constrained project scheduling problem rcpsp is presented given are n activities which have to be processed without preemptions during the processing period of an activity constant amounts of renewable resources are needed where the available capacity of each resource type is limited furthermore finish start precedence relations between the activities are given the objective is to determine a schedule with minimal makespan the lower bound calculations are based on two methods for proving infeasibility of a given threshold value t for the makespan the first uses constraint propagation techniques while the second is based on a linear programming formulation a column generation procedure is presented for the linear programming formulation and computational results are reported for an algorithm combining both concepts

adaptive selection of face coarse degrees of freedom in the bddc and the feti dp iterative substructuring methods we propose a class of method for the adaptive selection of the coarse space of the bddc and feti dp iterative substructuring methods the methods work by adding coarse degrees of freedom constructed from eigenvectors associated with intersections of selected pairs of adjacent substructures it is assumed that the starting coarse degrees of freedom are already sufficientto prevent relative rigid body motions in any selected pair of adjacent substructures a heuristic indicator of the the condition number is developed and a minimal number of coarse degrees of freedom is added to decrease the indicator under a given threshold it is shown numerically on 2d elasticity problems that the indicator based on pairs of substructures with common edges predicts the actual condition number reasonably well and that the method can select adaptively the hard part of the problem and concentrate computational work there to achieve good convergence of the iterations at a modest cost

lower bounds for resource constrained project scheduling problems abstract a destructive lower bound for the multi mode resource constrained project scheduling problem with minimal and maximal time lags is presented given are n activities which may be processed in different modes without preemptions during processing certain amounts of renewable and non renewable resources are needed where the available capacity of each resource type is limited furthermore minimal and maximal time lags between the activities are given the objective is to determine a schedule with minimal makespan the lower bound calculations are based on two methods for proving infeasibility of a given threshold value t for the makespan the first uses constraint propagation techniques while the second is based on a linear programming formulation which is solved by a column generation procedure computational results are reported for several test instances of the multi mode problem with and without time lags and the single mode version with time lags

projector preconditioning and domain decomposition methods we show that the method of preconditioning by conjugate projector may be exploited in the context of domain decomposition methods the performance of the method is assessed by model analysis of finite difference discretization of the poisson equation on a rectangular region

abstract robust coarse spaces for systems of pdes via generalized eigenproblems in the overlaps coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations pdes however it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the pde coefficients especially for systems of pdes here we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities we achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence we prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous pdes and systems of pdes that confirm this property

feti and neumann neumann iterative substructuring methods connections and new results the feti and neumann neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations they are iterative substructuring methods and have many algorithmic components in common but there are also differences the purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of feti algorithms bounds on the rate of convergence which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients are established for these new algorithms the theory for a variant of the neumann neumann algorithm is also redeveloped stressing similarities to that for the feti methods

a branch cut and price algorithm for the piecewise linear transportation problem abstract in this paper we present an exact solution method for the transportation problem with piecewise linear costs this problem is fundamental within supply chain management and is a straightforward extension of the fixed charge transportation problem we consider two dantzig wolfe reformulations and investigate their relative strength with respect to the linear programming lp relaxation both theoretical and practical through tests on a number of instances based on one of the proposed formulations we derive an exact method by branching and adding generalized upper bound constraints from violated cover inequalities the proposed solution method is tested on a set of randomly generated instances and compares favorably to solving the model using a standard formulation solved by a state of the art commercial solver

dual primal feti methods for linear elasticity dual primal feti methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations as in primal iterative substructuring methods while most of the constraints are enforced by lagrange multipliers as in one level feti methods these methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems the purpose of this article is to develop strategies for selecting these constraints which are enforced throughout the iterations such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them the algorithms are described in terms of a change of basis that has proven to be quite robust in practice a theoretical analysis is provided for the case of linear elasticity and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the young s modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain the strategies have already proven quite successful in large scale implementations of these iterative methods 2006 wiley periodicals inc

decomposing combinatorial auctions and set packing problems combinatorial auctions allow bidders to bid on bundles of items rather than just on single items the winner determination problem in combinatorial auctions is the problem of determining the allocation of items to bidders such that the sum of the accepted bid prices is maximized this problem is equivalent to the well known maximum weight set packing problem even though these problems are np hard in general they can be solved in polynomial time on instances whose associated item graphs have bounded treewidth called structured item graphs however the tractability of determining whether for a given problem instance a structured item graph of fixed treewidth exists and if so computing one efficiently was an open problem in this article we solve this problem by proving that deciding the existence of structured item graphs is computationally intractable even for treewidth 3 motivated by this unfavorable complexity result we investigate other structural restrictions and we show that the notion of hypertree decomposition a well studied measure of hypergraph cyclicity turns out to be most useful here indeed we show that the winner determination problem is solvable in polynomial time on instances whose dual auction hypergraphs have bounded hypertree width our solution method is based on encoding winner determination via a constraint satisfaction optimization problem and on exhibiting an algorithm to solve this latter problem efficiently for such structurally restricted instances the class of tractable instances identified by our approach while being efficiently recognizable properly contains the class of instances having a structured item graph moreover on the larger class our method solves winner determination with the same asymptotic complexity as the best algorithm proposed in the literature for the subclass of structured item graphs hypertree decompositions can equally profitably be applied to the maximum weight independent set problem which is the dual problem of maximum weight set packing

a note on bounding a class of linear programming problems including cutting stock problems in lp problems with an extremely large number of possible structural variables solution to optimality is not always required a bound can be used to terminate such lps such a bound is presented for a class of problem that includes most cutting stock formulations

the classification problem for s local torsion free abelian groups of finite rank abstract suppose that n 2 and that s t are sets of primes then the classification problem for the s local torsion free abelian groups of rank n is borel reducible to the classification problem for the t local torsion free abelian groups of rank n if and only if s t

analysis of feti methods for multiscale pdes in this paper we study a variant of the finite element tearing and interconnecting feti method which is suitable for elliptic pdes with highly heterogeneous multiscale coefficients x in particular coefficients with strong variation within subdomains and or jumps that are not aligned with the subdomain interfaces using energy minimisation and cut off arguments we can show rigorously that for an arbitrary positive coefficient function alpha in l infty omega the condition number of the preconditioned feti system can be bounded by c 1 log h h 2 where h is the subdomain diameter and h is the mesh size and where the function c depends only on the coefficient variation in the vicinity of subdomain interfaces in particular if alpha omega i varies only mildly in a layer i of width near the boundary of each of the subdomains i then c alpha mathcal o h eta 2 independent of the variation of in the remainder i i of each subdomain and independent of any jumps of across subdomain interfaces the quadratic dependence of c on h can be relaxed to a linear dependence under stronger assumptions on the behaviour of in the interior of the subdomains our theoretical findings are confirmed in numerical tests

a branch price and cut algorithm for multi mode resource leveling abstract we propose a new mixed integer programming formulation and solution algorithm for a multi mode resource constrained project scheduling problem with availability constraints calendars and the objective to minimize the resource availability cost our model exploits the problem structure and has an exponential number of variables which necessitates a column generation approach the linear programming relaxation is strengthened by adding valid inequalities that need to be carefully separated in order to show the desired effect integer optimal solutions are obtained by an exact state of the art branch price and cut algorithm classical time indexed mixed integer programming formulations for similar problems quite often fail already on instances with only 30 jobs depending on the network complexity and the total freedom of arranging jobs a reason is the typically very weak linear programming relaxation our model can be interpreted as a non trivial dantzig wolfe reformulation of a time indexed formulation in particular for larger instances our reformulation gives significantly tighter dual bounds enabling us to optimally solve instances with 50 multi mode jobs this outperforms the classical formulation by far

a two level overlapping schwarz method with energy minimizing multiscale coarse basis functions abstract a two level overlapping schwarz method is developed for second order elliptic problems with highly oscillatory and high contrast coefficients for which it is known that the standard coarse problem fails to give a robust preconditioner in this paper we develop energy minimizing multiscale finite element functions to form a more robust coarse problem first a local spectral problem is solved in each non overlapping coarse subdomain and dominant eigenfunctions are selected as auxiliary functions which are crucial for the high contrast case the required multiscale basis functions are then obtained by minimizing an energy subject to some orthogonality conditions with respect to the auxiliary functions due to an exponential decay property the minimization problem is solved locally on oversampling subdomains that are unions of a few coarse subdomains the coarse basis functions are therefore local and can be computed efficiently the resulting preconditioner is shown to be robust with respect to the contrast in the coefficients as well as the overlapping width in the subdomain partition numerical results are included to confirm the theory and show the performance

analysis of feti methods for multiscale pdes part ii interface variation in this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting feti method and a variant thereof all floating feti we consider a scalar elliptic equation in a two or three dimensional domain with a highly heterogeneous multiscale diffusion coefficient this coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains in other words the subdomain partitioning does not need to resolve any jumps in the coefficient under suitable assumptions we derive bounds for the condition numbers of one level and all floating feti that are robust with respect to strong variations in the contrast in the coefficient and that are explicit in some geometric parameters associated with the coefficient variation in particular robustness holds for face edge and vertex islands in high contrast media as a central tool we prove and use new weighted poincare and discrete sobolev type inequalities that are explicit in the weight our theoretical findings are confirmed in a series of numerical experiments

a simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems several domain decomposition methods with lagrange multipliers have been recently designed for solving iteratively large scale systems of finite element equations while these methods differ typically by implementational details they share in most cases the same substructure based preconditioners that were originally developed for the feti method the success of these preconditioners is due to the fact that for homogeneous structural mechanics problems they ensure a computational performance that scales with the problem size in this paper we address the suboptimal behaviour of these preconditioners in the presence of material and or discretization heterogeneities we propose a simple and virtually no cost extension of these preconditioners that exhibits scalability even for highly heterogeneous systems of equations we consider several intricate structural analysis problems and demonstrate numerically the optimal performance delivered by the new preconditioners for problems with discontinuities copyright 1999 john wiley sons ltd

dynamic aggregation of set partitioning constraints in column generation column generation is often used to solve problems involving set partitioning constraints such as vehicle routing and crew scheduling problems when these constraints are in large numbers and the columns have on average more than 8 12 nonzero elements column generation often becomes inefficient because solving the master problem requires very long solution times at each iteration due to high degeneracy to overcome this difficulty we introduce a dynamic constraint aggregation method that reduces the number of set partitioning constraints in the master problem by aggregating some of them according to an equivalence relation to guarantee optimality this equivalence relation is updated dynamically throughout the solution process tests on the linear relaxation of the simultaneous vehicle and crew scheduling problem in urban mass transit show that this method significantly reduces the size of the master problem degeneracy and solution times especially for larger problems in fact for an instance involving 1 600 set partitioning constraints the master problem solution time is reduced by a factor of 8

an adaptive multipreconditioned conjugate gradient algorithm this article introduces and analyzes a new adaptive algorithm for solving symmetric positive definite linear systems in cases where several preconditioners are available or the usual preconditioner is a sum of contributions a new theoretical result allows us to select at each iteration whether a classical preconditioned conjugate gradient cg iteration is sufficient i e the error decreases by a factor of at least some chosen ratio or whether convergence needs to be accelerated by performing an iteration of multipreconditioned cg 4 this is first presented in an abstract framework with the one strong assumption being that a bound for the smallest eigenvalue of the preconditioned operator is available then the algorithm is applied to the balancing domain decomposition method and its behavior is illustrated numerically in particular it is observed to be optimal in terms of local solves for both well conditioned and ill conditioned test cases which makes it a good candidate to be a default par

adaptive selection of primal constraints for isogeometric bddc deluxe preconditioners isogeometric analysis has been introduced as an alternative to finite element methods in order to simplify the integration of computer aided design cad software and the discretization of variational problems of continuum mechanics in contrast with the finite element case the basis functions of isogeometric analysis are often not nodal as a consequence there are fat interfaces which can easily lead to an increase in the number of interface variables after a decomposition of the parameter space into subdomains building on earlier work on the deluxe version of the bddc balancing domain decomposition by constraints family of domain decomposition algorithms several adaptive algorithms are developed in this paper for scalar elliptic problems in an effort to decrease the dimension of the global coarse component of these preconditioners numerical experiments provide evidence that this work can be successful yielding scalable and quasi optimal adaptive bddc algorithms for isogeometric discretizations

technical note an improved dual based algorithm for the generalized assignment problem the generalized assignment problem gap determines the minimum cost assignment of n jobs to m agents such that each job is assigned to exactly one agent subject to an agent s capacity existing solution algorithms have not solved problems with more than 100 decision variables this paper designs an optimization algorithm for the gap that effectively solves problems with up to 500 variables compared with existing procedures this algorithm requires fewer enumeration nodes and shorter running times improved performance stems from an enhanced lagrangian dual ascent procedure that solves a lagrangian dual at each enumeration node adding a surrogate constraint to the lagrangian relaxed model and an elaborate branch and bound scheme an empirical investigation of various problem structures not considered in existing literature is also presented

analysis of feti dp and bddc for linear elasticity in 3d with almost incompressible components and varying coefficients inside subdomains feti dp dual primal finite element tearing and interconnecting methods are nonoverlapping domain decomposition methods which are used to solve large algebraic systems of equations that arise e g from problems in linear elasticity good convergence bounds for problems of compressible linear elasticity are well known for two and three dimensional problems more recently feti dp and bddc balancing domain decomposition by constraints methods have been developed that are robust also in the regime of homogeneous almost incompressible linear elasticity the coarse space of such methods is large especially in 3d three dimensions and its implementation needs knowledge of geometrical information here the convergence of feti dp methods for problems in 3d with almost incompressible inclusions or compressible inclusions with different material parameters embedded in a compressible matrix material is analyzed for such problems where the material is compressible in the vicinity of the subdomain interface

a branch and price algorithm for the generalized assignment problem the generalized assignment problem examines the maximum profit assignment of jobs to agents such that each job is assigned to precisely one agent subject to capacity restrictions on the agents a new algorithm for the generalized assignment problem is presented that employs both column generation and branch and bound to obtain optimal integer solutions to a set partitioning formulation of the problem

linear programming based algorithms for preemptive and non preemptive rcpsp abstract in this paper the rcpsp resource constrained project scheduling problem is solved using a linear programming model each activity may or may not be preemptive each variable is associated to a subset of independent activities antichains the properties of the model are first investigated in particular conditions are given that allow a solution of the linear program to be a feasible schedule from these properties an algorithm based on neighbourhood search is derived one neighbour solution is obtained through one simplex pivoting if this pivoting preserves feasibility methods to get out of local minima are provided the solving methods are tested on the psplib instances in a preemptive setting and prove efficient they are used when preemption is forbidden with less success and this difference is discussed

measure reducibility of countable borel equivalence relations we show that every basis for the countable borel equivalence relations strictly above mathbb e 0 under measure reducibility is uncountable thereby ruling out natural generalizations of the glimm effros dichotomy we also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base using arguments substantially simpler than those previously employed

linear algebraic groups and countable borel equivalence relations this paper is a contribution to the study of borel equivalence relations on standard borel spaces i e polish spaces equipped with their borel structure in mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants frequently these objects can be viewed as elements of a standard borel space x and the equivalence turns out to be a borel equivalence relation e on x a complete classification of x up to e consists of finding a set of invariants i and a map c x i such that xey c x c y for this to be of any interest both i and c must be explicit or definable and as simple and concrete as possible the theory of borel equivalence relations studies the set theoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems

grasp for set packing problems abstract the principles of the greedy randomized adaptative search procedure grasp metaheuristic are instantiated for the set packing problem we investigated several construction phases and evaluated improvements based on advanced strategies these improvements include a self tuning procedure using reactive grasp an intensification procedure using path relinking and a procedure involving the diversification of the selection using a learning process two sets of various numerical instances were used to perform the computational experiments the first set contains randomly generated instances while the second includes instances relating to real problems in railway planning no metaheuristic has previously been applied to this combinatorial problem consequently we have discussed grasp s performances both in relation to lower upper bounds and to the results obtained with cplex when such results are available our analysis based on the average performances observed shows the impact of the suggested strategies and indicates the configuration that produces the best results

completeness of the isomorphism problem for separable c algebras this paper studies the descriptive set theoretical complexity of the isomorphism problem for separable c algebras we prove that the isomorphism problem for separable simple ai c algebras is complete in the class of orbit equivalence relations this means that any isomorphism problem arising from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple separable ai c algebras

min cut clustering we describe a decomposition framework and a column generation scheme for solving a min cut clustering problem the subproblem to generate additional columns is itself an np hard mixed integer programming problem we discuss strong valid inequalities for the subproblem and describe some efficient solution strategies computational results on compiler construction problems are reported

balancing neumann neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity balancing neumann neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients discretized with mixed finite or spectral elements with discontinuous pressures these domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions a polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case parallel and serial numerical experiments confirm the theoretical results indicate that they still hold for systems with discontinuous coefficients and show that our algorithm is scalable parallel and robust with respect to material heterogeneities the results on heterogeneous general problems are also supported in part by our theory

project scheduling under partially renewable resource constraints we consider a generalization of the classical resource constrained project scheduling problem we introduce so called partially renewable resources by assuming for each resource a capacity on subsets of periods the concept of partially renewable resources is a fundamental tool in order to make e g timetabling and shift scheduling aspects amenable to project scheduling in addition partially renewable resources serve to model complicated labor regulations furthermore they cover traditional renewable and nonrenewable resource constraints as special cases we consider makespan minimization as objective for the exact solution of the problem we employ a basic enumeration scheme in order to speed up convergence we formulate bounds which take into account future resource consumption of partially renewable resources moreover we generalize the serial scheduling scheme in order to get fast approximation methods a rigorous assessment of the procedures is provided by solving progen instances generated under a full factorial test design besides the well known problem parameters we employ additionally three parameters which control the generation of partially renewable resources

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

on the complexity of the isomorphism relation for finitely generated groups working within the framework of descriptive set theory we show that the isomorphism relation for finitely generated groups is a universal essentially countable borel equivalence relation we also prove the corresponding result for the conjugacy relation for subgroups of the free group on two generators the proofs are group theoretic and we refer to descriptive set theory only for the relevant definitions and for motivation for the results

automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms we introduce spectral coarse spaces for the balanced domain decomposition and the finite element tearing and interconnecting methods these coarse spaces are specifically designed for the two level methods to be scalable and robust with respect to the coefficients in the equation and the choice of the decomposition we achieve this by solving generalized eigenvalue problems on the interfaces between subdomains to identify the modes that slow down convergence theoretical bounds for the condition numbers of the preconditioned operators which depend only on a chosen threshold and the maximal number of neighbors of a subdomain are presented and proved for the finite element tearing and interconnecting method there are two versions of the two level method one based on the full dirichlet preconditioner and the other on the cheaper lumped preconditioner some numerical tests confirm these results copyright 2013 john wiley sons ltd

the isomorphism problem for torsion free abelian groups is analytic complete we prove that the isomorphism problem for torsion free abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy namely 1 complete

an effective branch and price algorithm for the preemptive resource constrained project scheduling problem based on minimal interval order enumeration in this paper we address the preemptive resource constrained project scheduling problem prcpsp prcpsp requires a partially ordered set of activities to be scheduled using limited renewable resources such that any activity can be interrupted and later resumed without penalty the objective is to minimize the project duration this paper proposes an effective branch and price algorithm for solving prcpsp based upon minimal interval order enumeration involving column generation as well as constraint propagation experiments conducted on various types of instances have given very satisfactory results our algorithm is able to solve to optimality the entire set of j30 bl and pack instances while satisfying the preemptive requirement furthermore this algorithm provides improved best known lower bounds for some of the j60 j90 and j120 instances in the non preemptive case rcpsp

bddc and feti dp preconditioners with adaptive coarse spaces for three dimensional elliptic problems with oscillatory and high contrast coefficients abstract bddc and feti dp algorithms are developed for three dimensional elliptic problems with adaptively enriched coarse components it is known that these enriched components are necessary in the development of robust preconditioners to form the adaptive coarse components carefully designed generalized eigenvalue problems are introduced for each faces and edges and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance t o l upper bounds for condition numbers of the preconditioned systems are shown to be c t o l with the constant c depending only on the maximum number of edges and faces per subdomain and the maximum number of subdomains sharing an edge numerical results are presented to test the robustness of the proposed approach

overlapping schwarz methods with adaptive coarse spaces for multiscale problems in 3d we propose two variants of the overlapping additive schwarz method for the finite element discretization of scalar elliptic problems in 3d with highly heterogeneous coefficients the methods are efficient and simple to construct using the abstract framework of the additive schwarz method and an idea of adaptive coarse spaces in one variant the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problems on the faces in the other variant it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and subdomain edges the functions that constitute the coarse spaces are chosen adaptively and they correspond to the eigenvalues that are smaller than a given threshold the convergence rate of the preconditioned conjugate gradients method in both cases is shown to be independent of the variations in the coefficients for the sufficient number of eigenfunctions in the coarse space numerical results are given to support the theory

project scheduling with resource constraints a branch and bound approach abstract this paper describes a branch and bound algorithm for project scheduling with resource constraints the algorithmis based on the idea of using disjunctive arcs for resolving conflicts that are created whenever sets of activities have to be scheduled whose total resource requirements exceed the resource availabilities in some periods four lower bounds are examined the first is a simple lower bound based on longest path computations the second and third bounds are derived from a relaxed integer programming formulation of the problem the second bound is based on the linear programming relaxation with the addition of cutting planes and the third bound is based on a lagrangean relaxation of the formulation this last relaxation involves a problem which is a generalization of the longest path computation and for which an efficient though not polynomial algorithm is given the fourth bound is based on the disjunctive arcs used to model the problem as a graph we report computational results on the performance of each bound on randomly generated problems involving up to 25 activities and 3 resources