the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

learning overcomplete representations in an overcomplete basis the number of basis vectors is greater than the dimensionality of the input and the representation of an input is not a unique combination of basis vectors overcomplete representations have been advocated because they have greater robustness in the presence of noise can be sparser and can have greater flexibility in matching structure in the data overcomplete codes have also been proposed as a model of some of the response properties of neurons in primary visual cortex previous work has focused on finding the best representation of a signal using a fixed overcomplete basis or dictionary we present an algorithm for learning an overcomplete basis by viewing it as probabilistic model of the observed data we show that overcomplete bases can yield a better approximation of the underlying statistical distribution of the data and can thus lead to greater coding efficiency this can be viewed as a generalization of the technique of independent component analysis and provides a method for bayesian reconstruction of signals in the presence of noise and for blind source separation when there are more sources than mixtures

a bias correction method for identification of linear dynamic errors in variables models this paper considers the problem of identifying linear systems where the input is observed in white noise but the output is observed in colored noise which also includes process disturbances an efficient method is developed which can perform unbiased parameter estimation without utilizing a prefilter the developed method is characterized by attractive features direct use of the observed data without prefiltering no need to evaluate autocorrelation functions for the input noise no need to identify a high order augmented system and provision of a direct unbiased estimate of the system parameters without parameter extraction computer simulations are presented to illustrate its superior performance including its significantly reduced computational complexity

reconstructing random media ii three dimensional media from two dimensional cuts we report on an investigation concerning the utilization of morphological information obtained from a two dimensional 2d slice thin section of a random medium to reconstruct the full three dimensional 3d medium we apply a procedure that we developed in an earlier paper that incorporates any set of statistical correlation functions to reconstruct a fontainebleau sandstone in three dimensions since we have available the experimentally determined 3d representation of the sandstone we can probe the extent to which intrinsically 3d information such as connectedness is captured in the reconstruction we considered reconstructing the sandstone using the two point probability function and lineal path function as obtained from 2d cuts cross sections of the sample the reconstructions are able to reproduce accurately certain 3d properties of the pore space such as the pore size distribution the mean survival time of a brownian particle and the fluid permeability the degree of connectedness of the pore space also compares remarkably well with the actual sandstone however not surprisingly visualization of the 3d pore structures reveals that the reconstructions are not perfect a more refined reconstruction can be produced by incorporating higher order information at the expense of greater computational cost finally we remark that our reconstruction study sheds lightmore on the nature of information contained in the employed correlation functions thinsp copyright ital 1998 ital the american physical society less

strongly polynomial algorithms for the high multiplicity scheduling problem a high multiplicity scheduling problem consists of many jobs which can be partitioned into relatively few groups where all the jobs within each group are identical polynomial and even strongly polynomial algorithms for the standard scheduling problem in which all jobs are assumed to be distinct become exponential for the corresponding high multiplicity problem in this paper we study various high multiplicity problems of scheduling unit time jobs on a single machine we provide strongly polynomial algorithms for constructing optimal schedules with respect to several measures of efficiency completion time lateness tardiness the number of tardy jobs and their weighted counterparts the algorithms require a number of operations that are polynomial in the number of groups rather than in the total number of jobs as a by product we identify a new family of nxn transportation problems which are solvable in o n log n time by a simple greedy algorithm

a polynomial algorithm for multiprocessor scheduling with two job lengths the following multiprocessor scheduling problem was motivated by scheduling maintenance periods for aircraft each maintenance period is a job and the maintenance facilities are machines in this context there are very few different types of maintenances performed so it is natural to consider the problem with only a small fixed number c of different types of jobs each job type has a processing time and each machine is available for the same length of time a machine can handle at most one job at a time all jobs are released at time zero there are no due dates or precedence constraints and preemption is not allowed the question is whether it is possible to finish all jobs we call this problem the multiprocessor scheduling problem with c job lengths mspc scheduling problems such as mspc where we can partition the jobs into relatively few types such that all jobs of a certain type are identical are often called high multiplicity problems high multiplicity problems are interesting because their input is very compact the input to mspc consists of only 2c 2 numbers for the case c 2 we present a polynomial time algorithm we show that this algorithm guarantees a schedule that uses the minimum possible number of different one machine schedules namely three further we extend this algorithm to the case of machine dependent deadlines uniform parallel machines to a multi parametric case that contains the case of unrelated parallel machines and to some related covering problems finally we give some counterexamples showing why our results do not extend to the case c 2

exact algorithm for minimising the number of setups in the one dimensional cutting stock problem the cutting stock problem is that of finding a cutting of stock material to meet demands for small pieces of prescribed dimensions while minimising the amount of waste because changing over from one cutting pattern to another involves significant setups an auxiliary problem is to minimise the number of different patterns that are used the pattern minimisation problem is significantly more complex but it is of great practical importance in this paper we propose an integer programming formulation for the problem that involves an exponential number of binary variables and associated columns each of which corresponds to selecting a fixed number of copies of a specific cutting pattern the integer program is solved using a column generation approach where the subproblem is a nonlinear integer program that can be decomposed into multiple bounded integer knapsack problems at each node of the branch and bound tree the linear programming relaxation of our formulation is made tighter by adding super additive inequalities branching rules are presented that yield a balanced tree incumbent solutions are obtained using a rounding heuristic the resulting branch and price and cut procedure is used to produce optimal or approximately optimal solutions for a set of real life problems

central limit theorem for sequential monte carlo methods and its application to bayesian inference the term sequential monte carlo methods or equivalently particle filters refers to a general class of iterative algorithms that performs monte carlo approximations of a given sequence of distributions of interest t we establish in this paper a central limit theorem for the monte carlo estimates produced by these computational methods this result holds under minimal assumptions on the distributions t and applies in a general framework which encompasses most of the sequential monte carlo methods that have been considered in the literature including the resample move algorithm of gilks and berzuini j r stat soc ser b stat methodol 63 2001 127 146 and the residual resampling scheme the corresponding asymptotic variances provide a convenient measurement of the precision of a given particle filter we study in particular in some typical examples of bayesian applications whether and at which rate these asymptotic variances diverge in time in order to assess the long term reliability of the considered algorithm

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

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

random search in the one dimensional cutting stock problem abstract the paper reports on a randomized approach to the one dimensional cutting stock problem in a sequential heuristic the next pattern to be processed is chosen according to the best outcome of 200 trials of random bin packing the quality of the solutions is discussed and the problem of multiple occurrence of equivalent solutions with different patterns is explained by the cross over technique known from genetic algorithms for one data instance with 27 demand lengths reported by haessler 1975 four different optimal solutions are given

small covering designs by branch and cut a branch and cut algorithm for finding covering designs is presented its originality resides in the use of isomorphism pruning of the enumeration tree a proof that no 4 10 5 1 covering design with less than 51 sets exists is obtained together with all non isomorphic 4 10 5 1 covering designs with 51 sets

decomposition approaches for solving the integer one dimensional cutting stock problem with different types of standard lengths abstract this paper considers the integer one dimensional cutting stock problem with different types of standard lengths and the objective of cost minimization decomposition approaches based on the well known column generation technique for exactly solving the continuous relaxation of the problem and specific methods for separately solving the arising residual problems are proposed for evaluating the performance of the different approaches more than 12 000 randomly generated test problems have been solved the computational experiments prove the effectiveness of one of the proposed decomposition procedures with respect to the computing effort and the quality of the solutions obtained

the modified integer round up property of the one dimensional cutting stock problem a linear integer minimization problem ip has the modified integer round up property mirup if the optimal value of any instance of ip is not greater than the optimal value of the corresponding lp relaxation problem rounded up plus one the aim of this paper is to investigate numerically whether the mirup holds for the one dimensional cutting stock problem the computational experiments carried out with a lot of randomly generated instances of the one dimensional cutting stock problem show that for all instances an integer solution fulfills the mirup moreover in most cases the optimal value equals the round up optimal value of the corresponding lp relaxation similarly the approach proposed to verify the mirup is usable as a new heuristic procedure for solving one dimensional cutting stock problems this heuristic always leads to very good solutions being optimal in the most cases

solving airline crew scheduling problems by branch and cut the crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline s fleets the problem is especially important today since costs for flying personnel of major u s carriers have grown and now often exceed 1 3 billion a year and are the second largest item next to fuel cost of the total operating cost of major u s carriers thus even small percentage savings amount to substantial dollar amounts we present a branch and cut approach to solving to proven optimality large set partitioning problems arising within the airline industry we first provide some background related to this important application and then describe the approach for solving representative problems in this problem class the branch and cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree search algorithm that uses automatic reformulation procedures heuristics and linear programming technology to assist in the solution numerical experiments are reported for a sample of 68 large scale real world crew scheduling problems these problems include both pure set partitioning problems and set partitioning problems with side constraints these base constraints represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality an interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground

continuous knapsack sets with divisible capacities we study two continuous knapsack sets mathcal y ge y and mathcal y le y with n n integer one unbounded continuous and m m bounded continuous variables in either ge or le form when the coefficients of the integer variables are integer and divisible we show in both cases that the convex hull is the intersection of the bound constraints and 2 m 2m polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable the latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the ge case we also provide an extended formulation for the le case it follows that given a specific objective function optimization over both mathcal y ge y and mathcal y le y can be carried out by solving m m polynomial size linear programs a further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 after scaling in any non trivial facet defining inequality of the convex hull of such sets

on the effectiveness of set covering formulations for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is one of the most important problems in distribution and transportation a classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem the method starts by solving its linear programming relaxation via column generation and then uses a branch and bound strategy to find an integer solution to the set covering problem a solution to the vrptw an empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely efficient in this paper we explain this behavior by demonstrating that for any distribution of service times time windows customer loads and locations the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases

strongly consistent coefficient estimate for errors in variables models for the single input single output siso errors in variables system it is assumed that the system input is an arma process and that the driven noise of the system input and the observation noise are jointly gaussian the two dimensional observation made on system input and output is represented as a two dimensional 2d arma system of minimum phase driven by a sequence of 2d i i d gaussian random vectors innovation representation it is shown that the resulting arma system is identifiable i e its coefficients are uniquely defined under reasonable conditions recursive algorithms are proposed for estimating coefficients of the arma representation including those contained in the original siso system the estimates are proved to be convergent to the true values with probability one and the convergence rate is derived as well the theoretical results are justified by numerical simulation

brief identification of dynamic errors in variables models approaches based on two dimensional arma modeling of the data in this paper we propose a parametric and a non parametric identification algorithm for dynamic errors in variables model we show that the two dimensional process composed of the input output data admits a finite order arma representation the non parametric method uses the arma structure to compute a consistent estimate of the joint spectrum of the input and the output a frisch scheme is then employed to extract an estimate of the joint spectrum of the noise free input output data which in turn is used to estimate the transfer function of the system the parametric method exploits the arma structure to give estimates of the system parameters the performances of the algorithms are illustrated using the results obtained from a numerical simulation study

on dantzig wolfe decomposition in integer programming and ways to perform branching in a branch and price algorithm dantzig wolfe decomposition as applied to an integer program is a specific form of problem reformulation that aims at providing a tighter linear programming relaxation bound the reformulation gives rise to an integer master problem whose typically large number of variables is dealt with implicitly by using an integer programming column generation procedure also known as branch and price algorithm there is a large class of integer programs that are well suited for this solution technique in this paper we propose to base the dantzig wolfe decomposition of an integer program on the discretization of the integer polyhedron associated with a subsystem of constraints as opposed to its convexification this allows us to formulate the integrality restriction directly on the master variables and sets a theoretical framework for dealing with specific issues such as branching or the introduction of cutting planes in the master we discuss specific branching schemes and their effect on the structure of the column generation subproblem we give theoretical bounds on the complexity of the separation process and the extent of the modifications to the column generation subproblem our computational tests on the cutting stock problem and a generalisation the cutting strip problem show that in practice all fractional solutions can be eliminated using branching rules that preserve the tractability of the subproblem but there is a trade off between branching efficiency and subproblem tractability

identification of errors in variables systems with arma observation noises abstract the purpose of the paper is to give the recursive estimate for coefficients of the single input single output siso errors in variables eiv system and to prove its strong consistency the observation noises of the eiv system are assumed to be arma processes and the system input is taken to be an iid sequence the recursive estimation algorithm is given with the help of stochastic approximation sa method and the strong consistency is established under reasonable conditions the numerical simulation is consistent with the theoretical analysis

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

polytopes associated with symmetry handling this paper investigates a polyhedral approach to handle symmetries in mixed binary programs we study symretopes i e the convex hulls of all binary vectors that are lexicographically maximal in their orbit with respect to the symmetry group these polytopes turn out to be quite complex for practical use we therefore develop an integer programming formulation with ternary coefficients based on knapsack polytopes arising from a single lexicographic order enforcing inequality we show that for these polytopes the optimization as well as the separation problem of minimal cover inequalities can be solved in almost linear time we demonstrate the usefulness of this approach by computational experiments showing that it is competitive with state of the art methods and is considerably faster for specific problem classes

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

a polynomial time opt 1 algorithm for the cutting stock problem with a constant number of object lengths in the cutting stock problem we are given a set of objects of different types and the goal is to pack them all in the minimum possible number of identical bins all objects have integer lengths and objects of different types have different sizes the total length of the objects packed in a bin cannot exceed the capacity of the bin in this paper we consider the version of the problem in which the number of different object types is constant and we present a polynomial time algorithm that computes a solution using at most one more bin than an optimum solution

robust full bayesian learning for radial basis networks we propose a hierarchical full bayesian model for radial basis networks this model treats the model dimension number of neurons model parameters regularization parameters and noise parameters as unknown random variables we develop a reversible jump markov chain monte carlo mcmc method to perform the bayesian computation we find that the results obtained using this method are not only better than the ones reported previously but also appear to be robust with respect to the prior specification in addition we propose a novel and computationally efficient reversible jump mcmc simulated annealing algorithm to optimize neural networks this algorithm enables us to maximize the joint posterior distribution of the network parameters and the number of basis function it performs a global search in the joint space of the parameters and number of parameters thereby surmounting the problem of local minima to a large extent we show that by calibrating the full hierarchical bayesian prior we can obtain the classical akaike information criterion bayesian information criterion and minimum description length model selection criteria within a penalized likelihood framework finally we present a geometric convergence theorem for the algorithm with homogeneous transition kernel and a convergence theorem for the reversible jump mcmc simulated annealing method

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

bidimensional packing by bilinear programming we consider geometric problems in which rectangles have to be packed into identical squares these problems turn out to be very hard in practice and ilp formulations in which variables specify the coordinates in the packing perform very poorly while most methods developed so far are based on simple geometric considerations a recent landmark result of fekete and schepers suggests to put these geometric aspects aside and use the most advanced tools for the one dimensional case in this paper we make additional progress in this direction especially on the basic question does a given set of rectangles fit into a square which turns out to be the bottleneck of all the approaches known given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles respectively in addition we illustrate how to solve the associated separation problem as a bilinear program for which we develop a solution method that turns out to be fast in practice and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature the same tools are used to derive lower bounds for the two dimensional bin packing problem corresponding to the determination of an optimal pair of so called dual feasible functions these lower bounds in many cases equal those obtained by the customary set covering formulation for which column generation is very hard but are computable within a time that is smaller by some orders of magnitude this allows us to close a few of the benchmark instances in the literature

atm vp based network design abstract we consider the problem of designing an atm vp based leased line backbone network given point to point communication demands having predefined sizes in a network the problem is to find configurations of demand routes and link facilities installed on each edge satisfying all demands at minimum cost under some constraints one of the most important constraints is that a single demand cannot be split over multiple link facilities this is a sort of bin packing constraint we propose an integer programming formulation of the problem and an algorithm to solve it an efficient column generation technique to solve the linear programming relaxation is proposed and a valid inequality is used to strengthen the integer programming formulation the algorithm incorporates the column generation technique and the cutting plane approach into a branch and bound scheme we test the proposed algorithm on some real problems the results show that the algorithm can be used to solve the problems within reasonably small computing times

improving discrete model representations via symmetry considerations in this paper we focus on a useful modeling concept that is frequently ignored while formu lating discrete optimization problems very often there exists a natural symmetry inherent in the problem itself that if propagated to the model can hopelessly mire a branch and bound solver by burdening it to explore and eliminate such alternative symmetric solutions we discuss three applications where such a symmetry arises a telecommunications network design problem a noise pollution problem and a machine procurement and operation problem for each case we identify the indistinguishable objects in the model that create the problem symmetry and show how imposing certain decision hierarchies within the model significantly enhances its solvability while using a popular modern day commercial branch and cut software cplex 6 5

computing in permutation and matrix groups ii backtrack algorithm this is the second paper in a series which discusses computation in permutation and matrix groups of very large order the essential aspects of a backtrack algorithm which searches these groups are presented we then uniformly describe algorithms for computing centralizers intersections and set stabilizers as well as an algorithm which determines whether two elements are conjugate

reformulations in mathematical programming automatic symmetry detection and exploitation if a mathematical program has many symmetric optima solving it via branch and bound techniques often yields search trees of disproportionate sizes thus finding and exploiting symmetries is an important task we propose a method for automatically finding the formulation group of any given mixed integer nonlinear program and for reformulating the problem by means of static symmetry breaking constraints the reformulated problem which is likely to have fewer symmetric optima can then be solved via standard branch and bound codes such as cplex for linear programs and couenne for nonlinear programs our computational results include formulation group tables for the miplib3 miplib2003 globallib and minlplib instance libraries and solution tables for some instances in the aforementioned libraries

extended formulations for packing and partitioning orbitopes we give compact extended formulations for the packing and partitioning orbitopes with respect to the full symmetric group described and analyzed in kaibel and pfetsch kaibel v m e pfetsch 2008 packing and partitioning orbitopes math programming ser a114 1 1 36 these polytopes are the convex hulls of all 0 1 matrices with lexicographically sorted columns and at most respectively exactly one 1 entry per row they are important objects for symmetry reduction in certain integer programs using the extended formulations we also derive a rather simple proof of the fact established in the paper mentioned above that basically shifted column inequalities suffice to describe those orbitopes linearly

computational study of a column generation algorithm for bin packing and cutting stock problems this paper reports on our attempt to design an efficient exact algorithm based on column generation for the cutting stock problem the main focus of the research is to study the extend to which standard branch and bound enhancement features such as variable fixing the tightening of the formulation with cutting planes early branching and rounding heuristics can be usefully incorporated in a branch and price algorithm we review and compare lower bounds for the cutting stock problem we propose a pseudo polynomial heuristic we discuss the implementation of the important features of the integer programming column generation algorithm and in particular the implementation of the branching scheme our computational results demonstrate the efficiency of the resulting algorithm for various classes of bin packing and cutting stock problems

polynomiality for bin packing with a constant number of item types we consider the bin packing problem with d different item sizes si and item multiplicities ai where all numbers are given in binary encoding this problem formulation is also known as the one dime

tighter relaxations for the cutting stock problem in the cutting stock problem csp a given order for smaller pieces has to be cut from larger stock material in such a way that the number of stock material needed is minimal based on the classical integer linear programming model the common solution technique consists of solving the corresponding continuous relaxation problem followed by several heuristics which construct integer solutions in many cases an optimal solution can be obtained quickly in this way but for instances which do not possess the integer round up property the optimality of the solution obtained cannot be verified by means of the lp bound in order to overcome this non satisfactory situation two tighter relaxations of the csp are proposed and results of theoretical and numerical investigations are presented

identification of errors in variables systems with nonlinear output observations this paper concerns the identification problem of errors in variables eiv systems with nonlinear output observations under independent and identically distributed iid gaussian inputs with unknown variance recursive algorithms for estimating the parameters of the eiv systems are presented for a large class of nonlinear observations conditions on the system are imposed to guarantee the almost sure convergence of the estimates some simulation examples are included to justify the theoretical results

computing in permutation and matrix groups i normal closure commutator subgroups series this paper is the first in a series which discusses computation in permutation and matrix groups of very large order the fundamental concepts are defined and some algorithms which perform elementary operations are presented algorithms to compute normal closures commutator subgroups derived series lower central series and upper central series are presented

integral decomposition in polyhedra we say that a polyhedronp satisfies weak integral decomposition if whenever an integral vector is the sum ofk vectors inp it is also the sum ofk integral vectors inp this property is related to rounding results for packing and covering problems we study the property and two related properties and give results concerning integral polymatroids totally unimodular matrices and network flows pairs of strongly base orderable matroids and branchings in directed graphs

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

pruning by isomorphism in branch and cut the paper presents a branch and cut for solving 0 1 integer linear programs having a large symmetry group the group is used for pruning the enumeration tree and for generating cuts the cuts are non standard cutting integer feasible solutions but leaving the optimal value of the problem unchanged pruning and cut generation are performed by backtracking procedures using a schreier sims table for representing the group applications to hard set covering problems and to the generation of covering designs and error correcting codes are presented

an asymptotically exact algorithm for the high multiplicity bin packing problem the bin packing problem consists of finding the minimum number of bins of given capacity d required to pack a set of objects each having a certain weight we consider the high multiplicity version of the problem in which there are only c different weight values we show that when c 2 the problem can be solved in time o log d for the general case we give an algorithm which provides a solution requiring at most c 2 bins more than the optimal solution i e an algorithm that is asymptotically exact for fixed c the complexity of the algorithm is o poly log d where poly is a polynomial function not depending on c

ilp approaches to the blockmodel problem blockmodelling is a method for identifying structural similarities or equivalences between elements which has applications in a variety of contexts including multiattribute performance assessment one criterion for forming blocks results in a difficult non linear integer programme we give several integer linear programming formulations of this problem and provide comparative computational results we show that methods of reducing symmetry proposed by sherali and smith are not effective in this case and propose an iterative approach in which the size of the problem is reduced

sequential monte carlo samplers summary we propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space each distribution being known up to a normalizing constant these probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential monte carlo methods this methodology allows us to derive simple algorithms to make parallel markov chain monte carlo algorithms interact to perform global optimization and sequential bayesian estimation and to compute ratios of normalizing constants we illustrate these algorithms for various integration tasks arising in the context of bayesian inference

parallel machine scheduling linear programming and parameter list scheduling heuristics in this paper we consider a class of parallel machine scheduling problems and their associated set partitioning formulations we show that the tightness of the linear programming relaxation of these formulations is directly related to the performance of a class of heuristics called parameter list scheduling heuristics this makes it possible to characterize the worst possible gap between optimal solutions for the scheduling problems and the corresponding linear programming relaxations in the case of the classical parallel machine weighted completion time model we also show that the solution to the linear programming relaxation of the set partitioning formulation is asymptotically optimal under mild assumptions on the distribution of job weights and processing times finally we extend most of the results to the time discretized formulation of machine scheduling problems

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

computing two dimensional integer hulls an optimal algorithm is presented for computing the smallest set of linear inequalities that define the integer hull of a possibly unbounded two dimensional convex polygon r input to the algorithm is a set of linear inequalities defining r and the integer hull computed is the convex hull of the integer points of r it is proven that the integer hull has at most o n log amax inequalities where n is the number of input inequalities and amax is the magnitude of the largest input coefficient it is shown that the algorithm presented has complexity o n log amax and that this is optimal by proving that the integer hull may have omega n log a max inequalities in the worst case

on the bin packing problem with a fixed number of object weights abstract we consider a bin packing problem where the number of different object weights is fixed to c we analyze a simple approximate approach and show that it leads to an asymptotically exact polynomial algorithm with absolute error 0 if c 2 at most 1 if 1 6 a consequence of our analysis is a new upper bound on the gap between the optimal value of the problem at hand and the round up of the optimal value of the linear relaxation of its gilmore gomory formulation

worst case analyses linear programming and the bin packing problem in this paper we consider the familiar bin packing problem and its associated set partitioning formulation we show that the optimal solution to the bin packing problem can be no larger than 4 3 zlp wherezlp is the optimal solution value of the linear programming relaxation of the set partitioning formulation an example is provided to show that the bound is tight a by product of our analysis is a new worst case bound on the performance of the well studied first fit decreasing and best fit decreasing heuristics

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

predicting the output from a complex computer code when fast approximations are available s we consider prediction and uncertainty analysis for complex computer codes which can be run at different levels of sophistication in particular we wish to improve efficiency by combining expensive runs of the most complex versions of the code with relatively cheap runs from one or more simpler approximations a bayesian approach is described in which prior beliefs about the codes are represented in terms of gaussian processes an example is presented using two versions of an oil reservoir simulator 1 c complex mathematical models implemented in large computer codes have been used to study real systems in many areas of scientific research sacks et al 1989 usually because physical experimentation is too costly and sometimes impossible as in the case of large environmental systems a computer experiment involves running the code with various input values for the purpose of learning something about the real system often a simulator can be run at different levels of complexity with versions ranging from the most sophisticated high level code to the most basic for example in 4 we consider two codes which simulate oil pressure at a well of a hydrocarbon reservoir both codes use finite element analysis in which the rocks comprising the reservoir are represented by small interacting grid blocks the flow of oil within the reservoir can be simulated by considering the interaction between the blocks the two codes differ in the resolution of the grid so that we have a very accurate slow version using many small blocks and a crude approximation using large blocks which runs much faster alternatively a mathematical model could be expanded to include more of the scientific laws underlying the physical processes simple fast versions of the code may well include the most important features and are useful for preliminary investigations in real time applications the number of runs from a high level simulator may be limited by expense then there is a need to trade off the complexity of the expensive code with the availability of the simpler approximations the purpose of the current paper is to explore ways in which runs from several levels of a code can be used to make inference about the output from the most complex code we may also have uncertainty about values for the input parameters which apply in any given application uncertainty analysis of computer codes describes how this uncertainty on the inputs affects our uncertainty about the output

survivable networks linear programming relaxations and the parsimonious property we consider the survivable network design problem the problem of designing at minimum cost a network with edge connectivity requirements as special cases this problem encompasses the steiner tree problem the traveling salesman problem and thek edge connected network design problem we establish a property referred to as the parsimonious property of the linear programming lp relaxation of a classical formulation for the problem the parsimonious property has numerous consequences for example we derive various structural properties of these lp relaxations we present some algorithmic improvements and we perform tight worst case analyses of two heuristics for the survivable network design problem

the integration of an interior point cutting plane method within a branch and price algorithm this paper presents a novel integration of interior point cutting plane methods within branch and price algorithms unlike the classical method columns are generated at a central dual solution by applying the analytic centre cutting plane method accpm on the dual of the full master problem first we introduce some modifications to accpm we propose a new procedure to recover primal feasibility after adding cuts and use for the first time a dual newton s method to calculate the new analytic centre after branching second we discuss the integration of accpm within the branch and price algorithm we detail the use of accpm as the search goes deep in the branch and bound tree making full utilization of past information as a warm start we exploit dual information from accpm to generate incumbent feasible solutions and to guide branching finally the overall approach is implemented and tested for the bin packing problem and the capacitated facility location problem with single sourcing we compare against cplex mip 7 5 as well as a classical branch and price algorithm

exploitation of sensitivity derivatives for improving sampling methods many application codes such as finite element structural analyses and computational fluid dynamics codes are capable of producing many sensitivity derivatives at a small fraction of the cost of the underlying analysis a simple variance reduction method is described that exploits such inexpensive sensitivity derivatives to increase the accuracy of sampling methods five examples including a finite element structural analysis of an aircraft wing are provided that illustrate an order of magnitude improvement in accuracy for both monte carlo and stratified sampling schemes

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

modified orbital branching for structured symmetry with an application to unit commitment the past decade has seen advances in general methods for symmetry breaking in mixed integer linear programming these methods are advantageous for general problems with general symmetry groups some important classes of mixed integer linear programming problems such as bin packing and graph coloring contain highly structured symmetry groups this observation has motivated the development of problem specific techniques in this paper we show how to strengthen orbital branching in order to exploit special structures in a problem s symmetry group the proposed technique to which we refer as modified orbital branching is able to solve problems with structured symmetry groups more efficiently one class of problems for which this technique is effective is when the solution variables can be expressed as 0 1 matrices where the problem s symmetry group contains all permutations of the columns we use the unit commitment problem an important problem in power systems to demonstrate the strength of modified orbital branching

packing d dimensional bins in d stages we consider the d dimensional bin packing problem the most relevant generalization of classical bin packing and show a general result about the asymptotic worst case ratio of a wide class of approximation algorithms that construct solutions in d stages containing many heuristics previously considered in the literature moreover we give the exact value of the asymptotic worst case ratio between the optimal solution and the best solution obtainable by packing the items in d stages showing how to achieve such a ratio efficiently the key property behind our result is the asymptotic optimality of the fractional relaxation of one dimensional bin packing an important by product of the approximation schemes for the problem from the 1980s which to the best of our knowledge is used here for the first time only for the sake of the analysis for the two dimensional case we push the approximablity threshold below 2 after more than 20 years reaching 1 691 but also improve and widely simplify the analysis of the previous best method from the 1980s moreover for d 3 we improve the previous approximability threshold by a factor of 1 691 a notable improvement when d is small

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

recent advances in the solution of quadratic assignment problems the quadratic assignment problem qap is notoriously difficult for exact solution methods in the past few years a number of long open qaps including those posed by steinberg 1961 nugent et al 1968 and krarup 1972 were solved to optimality for the first time the solution of these problems has utilized both new algorithms and novel computing structures we describe these developments as well as recent work which is likely to result in the solution of even more difficult instances

the sequential knapsack polytope in this paper we describe the convex hull of all solutions of the integer bounded knapsack problem in the special case when the weights of the items are divisible the corresponding inequalities are defined via an inductive scheme that can also be used in a more general setting

exploiting orbits in symmetric ilp this paper describes components of a branch and cut algorithm for solving integer linear programs having a large symmetry group it describes an isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group pruning and orbit computations are performed by backtracking procedures using a schreier sims table for representing the symmetry group applications to hard set covering problems generation of covering designs and error correcting codes are given

friendly bin packing instances without integer round up property it is well known that the gap between the optimal values of bin packing and fractional bin packing if the latter is rounded up to the closest integer is almost always null known counterexamples to this for integer input values involve fairly large numbers specifically the first one was derived in 1986 and involved a bin capacity of the order of a billion later in 1998 a counterexample with a bin capacity of the order of a million was found in this paper we show a large number of counterexamples with bin capacity of the order of a hundred showing that the gap may be positive even for numbers which arise in customary applications the associated instances are constructed starting from the petersen graph and using the fact that it is fractionally but not integrally 3 edge colorable

algorithms for highly symmetric linear and integer programs this paper deals with exploiting symmetry for solving linear and integer programming problems basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension

ranking lower bounds for the bin packing problem abstract in this paper we evaluate different known lower bounds for the bin packing problem including linear programming relaxation lp lagrangean relaxation lr lagrangean decomposition ld and the martello toth mt martello s toth p knapsack problems algorithms and computer implementations wiley new york 1990 lower bounds we give conditions under which lagrangean bounds are superior to the lp bound show that lagrangean decomposition ld yields the same bound as lagrangean relaxation lr and prove that the mt lower bound is a lagrangean bound evaluated at a finite set of lagrange multipliers hence it is no better than the lr and ld lower bounds

parallel machine scheduling with high multiplicity in high multiplicity scheduling problems identical jobs are encoded in the efficient format of describing one of the jobs and the number of identical jobs similarly identical machines are efficiently encoded in the same manner we investigate parallel machine high multiplicity problems where there are three possible machine speed structures identical proportional or unrelated for the objectives of minimizing the sum of job completion times and minimizing the makespan we consider both nonpreemptive and preemptive problems for some problems we develop polynomial time algorithms for several problems we demonstrate that the recognition versions can be solved in polynomial time while the optimization versions require pseudo polynomial time we also show that changing from standard binary encoding to high multiplicity encoding does not affect the complexity class of np complete problems

accurate uncertainty quantification using inaccurate computational models this paper proposes a novel uncertainty quantification framework for computationally demanding systems characterized by a large vector of non gaussian uncertainties it combines state of the art techniques in advanced monte carlo sampling with bayesian formulations the key departure from existing works is the use of inexpensive approximate computational models in a rigorous manner such models can readily be derived by coarsening the discretization size in the solution of the governing pdes increasing the time step when integration of odes is performed using fewer iterations if a nonlinear solver is employed or making use of lower order models it is shown that even in cases where the inaccurate models provide very poor approximations of the high fidelity response statistics of the latter can be quantified accurately with significant reductions in the computational effort multiple approximate models can be used and rigorous confidence bounds of the estimates produced are provided at all stages

dimension reduction for semidefinite programs via jordan algebras we propose a new method for simplifying semidefinite programs sdp inspired by symmetry reduction specifically we show if an orthogonal projection map satisfies certain invariance conditions restricting to its range yields an equivalent primal dual pair over a lower dimensional symmetric cone namely the cone of squares of a jordan subalgebra of symmetric matrices we present a simple algorithm for minimizing the rank of this projection and hence the dimension of this subalgebra we also show that minimizing rank optimizes the direct sum decomposition of the algebra into simple ideals yielding an optimal block diagonalization of the sdp finally we give combinatorial versions of our algorithm that execute at reduced computational cost and illustrate effectiveness of an implementation on examples through the theory of jordan algebras the proposed method easily extends to linear and second order cone programming and more generally symmetric cone optimization